{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear Algebra; Vectors and Matrices." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is a minimal notebook for learning practical linear algebra with Python. We take a visual and intuitive approach to illuminate some of the core ideas in linear algebra, enabled by computing. \n", "\n", "Linear algebra is a surprisingly useful subject, at the heart of computer graphics, cryptography, and machine learning. It is applied in data compression, game theory, and understanding networks. Engineering applications of linear algebra are everywhere: electric circuits, statics and dynamics, digital signal processing, optimization, robotics, multi-body dynamics… you name it!\n", "\n", "This learning module can be your launching pad to the wonderful world of _vector spaces_.\n", "\n", "Let's get started! We will be using our favorite libraries of the Python ecosystem: NumPy and Matplotlib. We also have a few helper functions in the `plot_helper.py` script, which will make it easy to visualize the ideas in these lessons. \n", "Go ahead and load these by executing the next two cells." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy\n", "from matplotlib import pyplot\n", "\n", "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import sys\n", "sys.path.append('../scripts/')\n", "\n", "# Our helper, with the functions: \n", "# plot_vector, plot_linear_transformation, plot_linear_transformations\n", "from plot_helper import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vectors" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### What's a vector?\n", "\n", "Vectors are everywhere: physics, engineering, mathematics, computer science, video games, and more. Each field's interpretation of what a vector *is* may be different, but vectors live a similar life in every space.\n", "\n", "The first episode in the wonderful video series, [_\"Essence of Linear Algebra\"_](http://3b1b.co/eola) tells you of three different ideas about vectors [1]:\n", "\n", "1. For physicists, a vector is an \"arrow\" of a given length (magnitude) and direction. It can represent directional quantities like velocity, force, acceleration.\n", "2. For computer scientists, a vector is an ordered list of numbers. It can represent a set of variables or features stored in order.\n", "3. For mathematicians, vectors are generic objects that behave a certain way when they are added or scaled: $\\mathbf{u}+\\mathbf{v}$, $\\alpha\\mathbf{v}$.\n", "\n", " \n", "#### How you think of a vector depends on who you are..." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In computer science and in data science, vectors are often multi-dimensional, that is, they have many components. They contain a set of ordered variables in a data model, like for example: the age, weight, daily hours of sleep, weekly hours of exercise, and blood pressure of an individual (five dimensions)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Visualizing vectors\n", "\n", "Let's start with the idea of a vector as an \"arrow\" (magnitude plus direction). We visualize a vector by placing this arrow with its tail at the origin of a coordinate system.\n", "But changing the position of the tail doesn't change the vector's magnitude or direction, so the vector is the same no matter where we draw it. \n", "\n", "In the code cell below, we define a list with a single vector of coordinates $(2, 2)$, and we use our custom function `plot_vector()` to plot the vector with its tail at four different positions on a 2D coordinate system. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "scrolled": false }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "vectors = [(2,2)]\n", "tails = [(0,0), (1,-3)]\n", "plot_vector(vectors, tails)\n", "pyplot.title(\"The same vector, with its tail at four locations.\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the 2D plane, we can see clearly the connection between the \"arrow\" idea of vector, and the \"list of numbers,\" which in this case represents the coordinates of the arrow head when the tail is at the origin of the coordinate system.\n", "\n", "The first coordinate designates the horizontal distance between head and tail, and the second coordinate designates the vertical distance between head and tail. We typically will denote horizontal and vertical axes as $x$ and $y$.\n", "\n", "In three dimensions, $x$ and $y$ are usually denoting the perpendicular axes on the horizontal plane, and the vertical axis is denoted by $z$. A 3D vector thus has three components: $(x, y, z)$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fundamental vector operations\n", "\n", "Two operations are the foundation of everything: **vector addition**, and **multiplication by a scalar** (i.e., scaling).\n", "\n", "Let's first visualize vector addition. Suppose we have two vectors: \n", "\n", "$$\n", " \\mathbf{a} = \\left[ \\begin{array}{c} -2 \\\\ 1 \\end{array} \\right], \\quad \n", " \\mathbf{b} = \\left[ \\begin{array}{c} 1 \\\\ -3 \\end{array} \\right] \n", "$$\n", "\n", "We can visualize vector addition as follows: draw vector $\\mathbf{a}$ with its tail at the origin; then draw vector $\\mathbf{b}$ with its tail on the head of $\\mathbf{a}$. If you now draw a vector from the origin to the head of $\\mathbf{b}$, that vector is $\\mathbf{a} + \\mathbf{b}$.\n", "\n", "With our helper function for plotting 2D vectors, it looks like this:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# vector addition\n", "a = numpy.array((-2,1))\n", "b = numpy.array((1,-3))\n", "origin = numpy.array((0,0))\n", "print()\n", "vectors = [a, b, a+b]\n", "tails = [origin, a, origin]\n", "plot_vector(vectors, tails)\n", "pyplot.title(\"Adding vectors with coordinates $(-2, 1)$ and $(1,-3)$.\\n\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this visualization of vector addition, the head of $\\mathbf{a} + \\mathbf{b}$ ends up at the coordinates resulting from adding the tail-to-head horizontal and vertical distances of $\\mathbf{a}$ and $\\mathbf{b}$. In other words, from adding the respective coordinates:\n", "\n", "$$\n", " \\left[ \\begin{array}{c} -2 \\\\ 1 \\end{array} \\right] + \n", " \\left[ \\begin{array}{c} 1 \\\\ -3 \\end{array} \\right] =\n", " \\left[ \\begin{array}{c} -2+1 \\\\ 1-3 \\end{array} \\right] = \n", " \\left[ \\begin{array}{c} -1 \\\\ -2 \\end{array} \\right]\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now look at multiplication by a scalar: essentially, the length of the vector is *scaled* by the scalar factor. If you multiply a vector by $2$, its length (magnitude) doubles. \n", "\n", "For example, if we scale by $2$ the vector $\\mathbf{c} = \\left[ \\begin{array}{c} 2 \\\\ 1 \\end{array} \\right]$, it looks like this:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# vector scaling\n", "c = numpy.array((2,1))\n", "vectors = [c, 2*c]\n", "plot_vector(vectors)\n", "pyplot.title(\"Scaling of the vector $(2,1)$ by the scalar $2$.\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The head of the vector $2\\mathbf{c}$ ends up at the coordinates resulting from scaling the tail-to-head horizontal and vertical distances of $\\mathbf{c}$:\n", "\n", "$$\n", " 2\\cdot\\left[ \\begin{array}{c} 2 \\\\ 1 \\end{array} \\right] =\n", " \\left[ \\begin{array}{c} 2\\cdot 2 \\\\ 2\\cdot 1 \\end{array} \\right] =\n", " \\left[ \\begin{array}{c} 4 \\\\ 2 \\end{array} \\right] \n", "$$" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "negative_scalar = -1\n", "negative_c = negative_scalar * c\n", "\n", "vectors = [c, negative_c]\n", "tails = [origin, origin]\n", "plot_vector(vectors, tails)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Basis vectors\n", "\n", "With the ideas of vector addition and multiplication by a scalar fresh in your mind, now imagine this. Any horizontal vector (i.e., having zero as its second component) can be scaled to have length $1$. \n", "\n", "For example, the vector $\\,\\mathbf{u} = \\left[ \\begin{array}{c} u \\\\ 0 \\end{array} \\right]$ scaled by $1/u$ becomes $\\left[ \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right]$.\n", "\n", "Similarly, any vertical vector (having zero as its first component) can be scaled to have length $1$.\n", "\n", "Going the opposite way, \n", "- scaling the vector $\\,\\mathbf{i}=\\left[ \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right]$ can give us all possible horizontal vectors, and \n", "- scaling the vector $\\,\\mathbf{j}=\\left[ \\begin{array}{c} 0 \\\\ 1 \\end{array} \\right]$ can give us all possible vertical vectors. \n", "\n", "Since every vector is the sum of a horizontal and a vertical one, it means we can generate *all vectors* by adding scaled versions of $\\mathbf{i}$ and $\\mathbf{j}$. That's why they are called **basis vectors**.\n", "\n", "For any vector, its components are the scalars we need to multiply the basis vectors by to generate it. For example:\n", "\n", "$$\n", " \\left[ \\begin{array}{c} 3 \\\\ 2 \\end{array} \\right] =\n", " 3\\cdot\\left[ \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right] +\n", " 2\\cdot\\left[ \\begin{array}{c} 0 \\\\ 1 \\end{array} \\right] =\n", " 3\\,\\mathbf{i} + 2\\,\\mathbf{j}\n", "$$\n", "\n", "Let's visualize this using our helper function." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The vector (3,2) as a linear combination of the basis vectors:\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# basis vector\n", "i = numpy.array((1,0))\n", "j = numpy.array((0,1))\n", "\n", "vec = 3*i + 2*j\n", "print(\"The vector (3,2) as a linear combination of the basis vectors:\")\n", "vectors = [i, j, 3*i, 2*j, vec]\n", "plot_vector(vectors)\n", "pyplot.title(\"The vector $(3,2)$ as a linear combination of the basis vectors.\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Linear combination and span\n", "\n", "Adding two vectors that were each multiplied by a scalar is called a **linear combination** of those two vectors. Thus, we say that every vector is some linear combination of the basis vectors.\n", "\n", "This brings us to the idea of the **span** of two vectors: the set of all possible linear combinations of the two. The second episode of the series [_\"Essence of Linear Algebra\"_](http://3b1b.co/eola) uses rich visuals to bring these ideas to life.\n", "\n", "\n", "In the code cells below, we will use the NumPy function [`randint`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.random.randint.html) to get random integers in an interval (in this case, from $-10$ to $10$). We then create a list of 30 random vectors on the plane via a linear combination of the basis vectors $\\mathbf{i}$ and $\\mathbf{j}$, and we draw them all. " ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "from numpy.random import randint" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# span\n", "vectors = []\n", "i = numpy.array((1,0))\n", "j = numpy.array((0,1))\n", "\n", "for _ in range(30):\n", " m = randint(-10,10)\n", " n = randint(-10,10)\n", " vectors.append(m*i + n*j)\n", " \n", "plot_vector(vectors)\n", "pyplot.title(\"Thirty random vectors, created from the basis vectors\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can imagine that if we created more and more random vectors in this way, eventually we will fill up the 2D plane. Indeed, the *span* of the basis vectors is the whole 2D space. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What if we tried the same experiment, but making linear combinations of the vectors $\\mathbf{a}$ and $\\mathbf{b}$, defined above?" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "vectors = []\n", "for _ in range(30):\n", " m = randint(-10,10)\n", " n = randint(-10,10)\n", " vectors.append(m*a + n*b)\n", " \n", "plot_vector(vectors)\n", "pyplot.title(\"Thirty random vectors, created as linear combinations of $\\\\mathbf{a}$ and $\\\\mathbf{b}$.\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In fact, we can *still* fill up the whole plane with infinite linear combinations of $\\mathbf{a}$ and $\\mathbf{b}$—they span the full 2D space. We're not forced to use the unit vectors $\\mathbf{i}$ and $\\mathbf{j}$ as our basis vectors: other pairs of vectors could form a basis. With $\\mathbf{i}$ and $\\mathbf{j}$, we saw that the components of a vector $\\mathbf{v}$ are the scalars needed in its corresponding linear combination of the basis vectors. If we were to use another pair of vectors as basis, we would need a different pair of scalars in the linear combination to get the same vector: we are _changing the coordinate system_." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's see another situation... we'll make linear combinations of the vector $\\mathbf{a}$, and a new vector, $\\mathbf{d} = \\left[ \\begin{array}{c} -1 \\\\ 0.5 \\end{array} \\right]$," ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": 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cuGGCaokaDoYDmdTmzZsxbdo0HD16FCdPnsSyZcvw2GOP4dy5c1Kf/fv3Y+nSpTh8+DBOnTqF3r17Y9y4cdBoNCasnKh+YziQSa1YsQIzZsyQHoeEhECj0eDy5ctafWbNmgU3NzcAwLx583D+/HkcOnTI6PUSNRT8M1AyqcDAQOnn4uJihIeHw9vbG6GhoQCA7OxsxMXFYdGiRVI/Z2dndOnSBceOHcO4ceMM2k5qaqre9vT09GpUT1R/WUw4VPaHJWq1GqWlpVCr1UaqqPZZ+hhqUv9rr72GyMhIdO3aFV9//TWEEMjJyZEuLzk5OWnNAVdXVyQlJRn8B0ceHh4G15KXl2eRf8jUkOePOahK/Y6OjkaoqGYsJhxiY2P1tms0GuTn5yMxMRFWVpZ5tczSx1CT+h9//HFMnz4dERERCAkJwaZNm+Di4oL4+HgAQEpKCmxtbaX+BQUFyMnJqXReVEdycjJUKlWtr7euNeT5Yw6qUn9ISIhxiqoBiwmH8pcfdFGr1UhISIC3tzfs7e2NVFXtsvQx1Eb9vXr1gq+vL3755Re8//770ousU6dOWnNAqVSiefPmlc6LMhcuXNDbnpGRgSFDhgAAPD094enpWa36TYnzx7Qsvf4HWUw4GHIaZm1tDXt7e4s4ZauIpY+hqvUXFRVpnREAQJcuXXD58mU4Ojqie/fuAACVSqW1zqysLISGhhq8na5du+ptd3BwkH62s7NrMPvf3LB+82F5525UrwQEBMiWpaeno3Xr1gCAZs2aoWfPnoiJiZHaVSoVkpKSMHz4cKPVSdTQMBzIpBITE3Hw4EHp8Z49e3Dp0iXMmjVLWrZ48WJEREQgKysLALB+/Xr4+vpizJgxRq+XqKGwmMtKVD99/PHHWLFiBVatWoXS0lIoFAp89913GDBggNRn0qRJyMzMxMiRI6FUKtGsWTNERUVZ5JuWRJaC4UAm9dJLL+Gll16qtN/s2bMxe/ZsI1RERAAvKxERkQ4MByIikmE4EBGRDMOBiIhkGA5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgMREckwHIiISIbhQEREMgwHIiKSYTgQEZEMw4GIiGQYDkREJMNwICIiGYYDERHJMByIiEiG4UBERDIMByIikmE4EBGRDMOBiIhkGA5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOZHL79u3DiBEjMGzYMAQFBWHy5MlISUnR6rN161YEBASgf//+GDt2LG7cuGGiaokaBoYDmdzMmTPxxhtv4Pjx4/j999/h6OiIUaNGoaCgAACwf/9+LF26FIcPH8apU6fQu3dvjBs3DhqNxsSVE9VfDAcyuYcffhgjRowAAFhZWWHu3LlITk5GXFwcAGDFihWYNWsW3NzcAADz5s3D+fPncejQIZPVTFTf2Zi6AKLIyEitx0qlEgBQVFSE7OxsxMXFYdGiRVK7s7MzunTpgmPHjmHcuHEGbSM1NVVve3p6ehWrJqrfLCYccnJy9Lar1WqUlpZCrVYbqaLaZ+ljqK36T5w4gVatWsHPzw/nzp0DADg5OWnNAVdXVyQlJVU6L8p4eHgYvP28vDyD12tOOH9Mqyr1Ozo6GqGimrGYcIiNjdXbrtFokJ+fj8TERFhZWebVMksfQ23UX1RUhA8++AAvvPAC4uPjER8fDwBISUmBra2t1K+goAA5OTmVzovqSE5OhkqlqvX11jXOH9OqSv0hISHGKaoGLCYcAgMD9bar1WokJCTA29sb9vb2Rqqqdln6GGqj/tmzZ+PRRx/Fa6+9BgDSi6xTp05ac0CpVKJ58+aVzosyFy5c0NuekZGBIUOGAAA8PT3h6elZnfJNivPHtCy9/gdZTDgYchpmbW0Ne3t7izhlq4ilj6Em9S9cuBBNmjRBeHg4FAoFAKB79+4AAJVKpbXOrKwshIaGGrydrl276m13cHCQfrazs2uQ+98csH7zYXnnblQvrV69GlevXsW2bdugUCgQGxuL2NhYNGvWDD179kRMTIzUV6VSISkpCcOHDzdhxUT1G8OBTG7Lli3YvXs35s2bh7i4OMTExCAqKkp6M3rx4sWIiIhAVlYWAGD9+vXw9fXFmDFjTFk2Ub1mMZeVqH7KycnBnDlzoNFo0K9fP622HTt2AAAmTZqEzMxMjBw5EkqlEs2aNUNUVJRFvmlJZCkYDmRSjo6OKC0trbTf7NmzMXv2bCNUREQALysREZEODAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgMREckwHIiISIbhQEREMgwHIiKSYTgQEZEMw4GIiGQYDkREJMNwICIiGYYDERHJMByIiEiG4UBERDIMByIikmE4EBGRDMOBiIhkGA5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgMREckwHIiISIbhQEREMgwHIiKSYTgQEZEMw4GIiGQYDmRyRUVFWLRoEWxsbHD16lVZ+9atWxEQEID+/ftj7NixuHHjhvGLJGpgGA5kUlevXsXgwYORlpaG0tJSWfv+/fuxdOlSHD58GKdOnULv3r0xbtw4aDQaE1RL1HAwHMikcnNzsXv3bjz11FM621esWIFZs2bBzc0NADBv3jycP38ehw4dMmaZRA2OjakLoIbN19cXAJCamipry87ORlxcHBYtWiQtc3Z2RpcuXXDs2DGMGzfO4O3oWn956enpBq+LqCGwmHDIycnR265Wq1FaWgq1Wm2kimqfpY+hJvXn5eUBuH8mUXasz507BwBwcnLSOv6urq5ISkqqdE6U5+HhUaVaqrJuc9GQ5485qEr9jo6ORqioZiwmHGJjY/W2azQa5OfnIzExEVZWlnm1zNLHUJP6k5KSAADnz5/HrVu3AADx8fEAgJSUFNja2kp9CwoKkJOTU+mcqK7k5GSoVKo6WXddasjzxxxUpf6QkBDjFFUDFhMOgYGBetvVajUSEhLg7e0Ne3t7I1VVuyx9DDWpv+zMwdfXF+3btwcA6QXWqVMnreOvVCrRvHnzSudEeRcuXNDbnpGRgSFDhgAAPD094enpWaX6zUFDnj/mwNLrf5DFhIMhp2HW1tawt7e3iFO2ilj6GKpbv52dHQDAwcFBem737t0BACqVSmt9WVlZCA0NrdI2unbtqrfdwcFBq5aGtv/NBes3H5Z37kYNRrNmzdCzZ0/ExMRIy1QqFZKSkjB8+HATVkZU/zEcyKwtXrwYERERyMrKAgCsX78evr6+GDNmjIkrI6rfLOayEtVPRUVFGDFiBO7evQsAmD59Ojw8PBAZGQkAmDRpEjIzMzFy5EgolUo0a9YMUVFRFvmGJZElYTiQSdna2iI6Olpvn9mzZ2P27NnGKYiIADAc6gUhBLZ89zNOnruM7p3aoHvHNvDt2Abt3JtDoVCYujwiskAMh3pAoVBg9oTBuJpxG4v+9Y203NFOCd+OraWw6N6pDXw7tkYLZ8u/k4KI6hbDoZ5QKBRY9fwkONs3wdvbvwUA5OQV4LeEFPyWkCL1s21kgz1vPY2pIb1MVCkRWQKGQz2iUCjw1swxcLJX4qX1e2Xtto1s8NOHr6F/984mqI6ILAlv+aiH5k4cip0LnoSVlfb7DUXFJZi6fCvWf30c+YVFJqqOiCwBw6GemjWqHyKXvoBGNtYAgCaNG0GhUCD99j3M2/hvtHj4NUxdugVHzyTgjsoyP+iMiOoOLyvVY5MGBeD7f87FI0s+Qe9uHbFp3gwsi4hCZHQs8gqL8NUvcfjqlzgAQCsXZ3SX3rS+f8eTd4dWaNLYtpKtEFF9xDOHem5EkA+Ohr8KZ/sm8O7QGvuWvoA/Pl2C9u4uWv3Sb9/DjzGJ+HDfUTy1eieWRUQhv7DYRFUTkakxHBqA/t0747P5T0qPezzkgZQvVuDRIbrvWLK1sYa90hbbD53E4dPnkZqVDSGEkaolInPAy0oNRHMn7Y8QtrKywpdLnkOHli2w+svDWm1FJaX494kY/PvE3x9419TBDqODffHpm2GwUzY2Ss1EZDo8c2jAyv42YuVzEyvt27OzB9a//CiDgaiBYDgQFs4YjU3zZgAAFs8cA3sdAXDij0sY9tpa3Mi6Y+zyiMgEGA4EAHjxkRDsfutpdG3XCr9uXAD3Zk6yPvEpN9B22kL0eGY5Q4KonmM4kGRmaB9MGxIIv4fa4peP34SHWzMAwOtTQ2FX7pZWhgRR/cdwIC2NbO7fo9DFwx0n18+HZ1s3hI3oA/Xhjdjw8nSGBFEDwXCgCrVzd8F/Pp4P9+b3LzHNnThUb0j0e3kNsu7xr62J6gOGA+nl3twJLZs7ay2rKCTOX03HtHUH0O/lNTyTILJwDAeqNn0h0XbaQrSdOh/fnTqLnLwCE1ZJRNXBP4KjGps7cSjmThyKD7/8AW99dgBFJRoAwI1bd/Hw4s0AgLauTeHfuZ3Wlw91bdcSto04BYnMEV+ZVGueHzcAga2a4MyNPCz67DuUajRSW2rWXaRm3cX3v8UDAPr7PoS97zyPtq7NTFUuEenBy0pU62ZPGIiio58gxL9LhX1Ss7Lx0vovEb73CBKu3DBidURkCIYD1QkrKyv8tPZ1LJ01Tmf7Xzfv4NuTf2D+1q/h+/RyWA19AW2nzse1m7eNXCkR6cJwoDqjUCiw7MkJWPvi1Er7KhvZIHLZC2j3wEeJE5FpMByozr06NRT/eiMMCoUCjw8P1tknv6gYQ179EDGXrhi5OiLShW9Ik1E8O3YgHJsocVulxrSQXpi4ZDM0D3xHRGFxCYJmr0Qja2scWjkHw4N8TVQtETEcyGgeHRqEouIS2DaywfG1ryH0jY9QUqpBt3atcOFautSvuLQUofPXAwCCvNqhf3dP6fZXnw6tYd+EHxtOVNd4WYmMquzvGkL8vfDbpoVo3MgGL4wfCHFiG6YMDpD1P3PpGtZ9dRzPhO9C7xdXwnHsy1j31TFjl03U4DAcyGR6eXXA/7YvQaBXewBA5LLZECe2oZdXuwqfM6ynFzzbuOHazdv86lKiOsTLSmRSXh6t4OWhvezMlsX4+udYTFm2Vdb/WNxFHIu7CABwslfCt0Mb+HZsjelDgzCkZ1djlEzUIPDMgczS5MGBOLx6Hpo0blRhH5W6AL8m/AmNRqCvz0NGrI6o/uOZA5mtkcE++DH8FYxdtAFOdk1wNzcPufmFsn6fHjqJ73+PR1/vh+DboTW6d7r/5rVnWzfYWFuboHIiy8czBzJrA7p74sTa19HKxRl/7V2F4K4ddPbLuK3CN/85i/d2H8S05dswbflW/Hkjy7jFEtUjDAcyewFd2uPA+y+iuZM9jn34Gob09AIA2NpUfFbQxsUZx+Mu4pf/JSE7h19ARFRVvKxEFqGVS1MAgKOdEodWvYypy7bCxtoKwV074q1Pv5H1PxJzAUdiLkiPW7doit7dOmLb6zPRwtnRWGUTWSyGA1kcpW0j7H93Nr76OQ6PDQuGk50Sc9d/qfc5CgXw3lMPMxiIDMTLSmSRGtnY4LFh9z+nac7EIdi16ClYW1lh9oRBsLZSyPrfyLoL36eX4Zv/xBq5UiLLxHAgi/DNN9+gV69eGDhwIAYPHoyEhASt9rARfRG57AW0d3dB4s534eGm+0uEJr2zFYohzzMkiCrBcCCzd/r0aTzxxBP4/PPP8Z///AfPPPMMRo4ciZycHK1+Ewf2xCtThqOLhztOrp8Pz7ZuAIBWzeWXkhgSRPoxHMjsrV69GmPGjIGX1/27lGbOnImSkhJERETI+ipt7//RXDt3F/zn4/nw69QWm18NgzixDV093GX9y0Lih9/jpWV7T8SgpLS0jkZDZBnM/g3pkpISZGRkVNovNzcXWVlZuHHjBhwcHIxQWe2z9DHUVf1Hjx7Fyy+/jNTUVGmZt7c3oqKi8Mgjj+h97udvTsetezk4c+YMdr00AQAwdcVO/HUzW6vf8yu2ST/vO3oKp5OuY8NL09HYthE0Gg2srMz/9yjOH9Oqav0tW7aEjY35/hesEGb+6WWpqanw8PCovCMRkQW5fv062rZta+oyKmT+vw4REZHRmf2ZQ9llpdzc3Ar7ZGVlYdq0aQCAiIgItGtX8Uc+mytLH0Nd1Z+dnY1+/fph9erVmDBhgrT87bffxvnz53HgwAGD1lPZpcmkpCS88MIL9x/0fBhobKez33tPj8eUgT2x7+c4TNPx/ROmwvljWlWt38HBwewvK5lvZf/Pxsam0lMvBwcH6cXfrl07dO1qeR/dbOljqMv6nZ2dYWVlpbXOgoICdOvWzeDtVNbv6q1ydz41tgOUuq8ZL/niBOycXbA26nf8+mcmDn/wikHbr2ucP6Zl6fXrwstKZPaGDh2KmJgY6bEQAnFxcRg+fHitbcOnfUvp5xfH9sUzYwZU2Pf1zV9Bpc7HkTOJ6DtnJTQaTa3VQWQuGA5k9hYuXIhDhw4hKSkJAPD555/D2toas2bNqpPtTR3QHa9PC4WLU8V3nJRdi/1v4hX0eOZdlJSU1EktRKZi9peViIKDgxEREYEZM2agSZMmsLKywpEjR+DoWHefk3RblYtpIYE4e/kaUtJu4bZKjdIKzhDOX02D28TXkPb1GihtbeusJiJjYjiQRZg4cSImTpxotO0N6O6JAd09tZZdTc/C+Lc34fyVNFn/7NwCNBk5FyueeRhvzRwLADgak4C9P8Vg+/y6OcMhqku8rERkoPX7TyAl/ZbePm9vPwDFkOexLvII7ubm47MfTmHikk+MVCFR7eGZA5GB1s6ZhrVzpqGoqAT/OZeMH2MSEXPpKn46e0nW99VPvpZ+/vbkHwh5ZQ2i171hzHKJaoThQFRFtrY2GBbYDcMCu2ktdx77ElR58u+4BoCf/5eEgOfeQ9y/lhijRKIaM/s/gjNUTk4OYmNjERgYWKdvVNYlSx+DJddf/mNaLly4UKP71JuNexl31QU621ydHZD+dTisrSv+itPqsuT9D7B+c8P3HIhq2cuTK/77i6x7ubAZ/g/sOPSLESsiqjqGA1EtyspW4Yvjp6WPDq/I0+F7oBjyPHYdOWWkyoiqhuFAVItcmzkhec/7yD+yCeLENqTuW4V/PvsIBvfoAvmXlwKzVkUwJMgsMRyI6lAb1+ZY9PgYRK97A5oT2yBObIOHa1NZP4YEmRuGA5GRXdv3QbVCoqiYH9FBxsNwIDKRqobEjPc/xa17ObK+RHWB4UBkYoaGRMylvzB43hrcyMqWr4SoljEciMxEZSHx183bSPwrHQPnhSMlLQtpt+5i+c4o4xdKDQLDgcjM6AsJALiSfgs9n1mG81duYFlEFOZt2GvcAqlBsPhwKCwsxNKlSzF69Gi8/vrrGDBgACZOnIiUlBStfvfu3UNYWBiCg4MREBCA5cuXwxz+OPzOnTtYtmwZBgwYgDFjxuDZZ5/FBx98IPt+AHOtv0xycjKGDx+OV155RWe7udf/ww8/SD+HhYUhISHBhNXcVxYSbk3lf22rKijGyPkfAwDW7/8Js1Z+BgC4cScHBUVF2Lp1KwICAtC/f3+MHTsWN27cMGrt+uzbtw8jRozAsGHDEBQUhMmTJ8ter+Za/4EDBzBu3DiEhoZiwIABCAwMxL59+2T9zLX+KhEWLj09XbRq1UpcvnxZnDhxQty9e1dMnTpV9OrVS6vf+PHjxZNPPimEEEKtVgsfHx+xdu1aU5SsZffu3cLb21vcvXtXqFQqERkZKdzc3MSSJUu0+plr/UIIsWvXLtGnTx/Rp08f0aNHD6FSqWR9zLn+33//XdjZ2Qnc/w4fsXLlStGmTRud4zC2iMO/CrtRcwRCnqv035j568SclZtFszFzhGurNuLmzZtCCCGWL18u/P39RWlpqYlHc1+jRo3EkSNHhBBClJaWilmzZglPT0+RmZkpTpw4Ifbs2SPc3d3Nsv6RI0eKiIgI6fF3330nrKysRHx8vFCpVGZff1VYfDgUFhaKuLg46cCoVCqxfv164eTkJPWJj48XAERiYqK0bNOmTcLNzc3kB+zQoUNi+/btQgghjeG5554Tnp6eUh9zrl8IIQ4ePCgKCwvFjBkzdIaDudc/adIkMW7cOCkcEhIShLu7u9iwYYOpSxOZ2SrxU9wFsf7r4+K58F2i75yVegOi2Zi5AiHPCeWw58XNO/eEEELcvXtX2NjYiKioKBOP5r4pU6ZoPT5z5owAII4ePSpOnDgh/P39xfz586V2c6o/JiZGFBcXS49VKpUAIPbv3y+9fs25/qqw+MtKtra26Nmzp/Q4LS0NERERmDdvnrTs2LFjcHBwQLduf3+KZlBQEDIzMxEfH2/Ueh80evRoPP3001rLlEolioqKpMfmXD8AjBkzBrZ6vgHN3Os/fvw4evToIT22srJCYGAgjh07ZsKq7nNt6oghPbvipUlDse2NMPy6cSEOrXwJjRvp/kDl7P//VNiCUoHOMxfj2s3b+CH2Etp18zeL8QBAZGSk1mOlUgkAKCoqQk5ODv744w8EBQVJ7c7OzujSpYtZ1B8YGAgbm/v7vri4GOHh4fD29kZoaCgAmH39VWHx4VAmLS0Nzz//PHr06IGRI0fi3XffldpSUlLg7u6u1b9ly5ZSm7k5ffo0pk6dKj22tPofZM713759G/fu3YObm5vW8pYtW5q8Nl0Kioqw/puf0MzBDtZWuj6Q4285eQXwnLkE/7ucimsufohNvm6kKqvmt99+Q+vWrdGnTx+kpd3/lr2y+VHG3I7HnDlz4OrqiuPHj+PIkSNwcLj/feOWUr8h6k04tG7dGtu2bUN8fDx+/PFHPPfcc1JbXl4eGjdurNW/7HFeXp5R66xMXFwcUlNTsXjxYmmZJdWviznXX7b9B898GjdubPLadFHa2uKH1fOQvn8Nji15DDi1G6ueGoPHhgbp7F9UUoJVXx5GCazxW14znDh70cgV61dYWIjw8HCsX78ejRo1QmHh/TMfXfPFnI7Hpk2bcPv2bQwbNgz9+/dHeno6AFhM/YYw23BYtmwZFAqF3n8xMTGy57Vq1QorV67Ep59+Kt1xYmdnJx20MmWP7ezszKb+tLQ0fPTRR/jyyy/h7OwsLbeU+itiivoNVbb98pfxgPv1mbo2gxQXYIBPRxz8/VylXUthhRFvfISoX/9nhMIM88ILL2DKlCmYPHkygL//U9U1X8zteFhbW2PZsmUQQmDt2rUALKv+ypjtN8G98cYbmD17tt4+LVq0QGlpqWy5l5cXACAxMRE+Pj7o1KkTbt68qdUnIyMDANCpU6daqlibofWXuXPnDqZPn45XXnkF/v7+Wv0soX59TFG/oVxcXODs7IzMzEyt5RkZGSavzVBtXJxx+9u1OHrmPHZ8+yMiD0ejRbvOuK0ulN0uXKIRmPD2Jrz39AQsDhtnoorvW7hwIWxsbLBixQppWevWrQH8PT/KZGRkSNf1TamoqEjrLNPKygqenp5ITEwEYP71V4XZnjk4ODigZcuWev/Z2Nhg9+7d+Oijj7SeW3aKV3aghg0bhtzcXFy8+PcpdUxMDNzc3ODn52fS+oH7b2KNHz8eCxYsQGBgIABg27Zt0rrMvf7KmKL+qhg6dKjWG+NCCMTFxWH48Iq/tMfc2NjYYIDvQ3hxdDB6KNIwq1sTvDRxSIX9l3z2HRoNm42YS1eMWOXfVq9ejatXr2Lbtm1QKBSIjY3F2bNn4ejoiB49emidlapUKiQlJZnF8QgICJAtS09Pl/6vMff6q8Jsw6EqPvvsM9y+fRsAUFBQgPfeew++vr7SHQN+fn4YP348wsPDAQD5+fnYvHkzFixYACsr0+6CgoICTJgwAX369EGrVq1w6dIlxMXFYevWrVIfc67fEOZe/8KFC/HTTz9Jj6OiomBtbY1Zs2aZsKrqe/PNN7Hry31o38IJL00cio7NGsNaUyzrV6LRIGj2SihHvGjUkNiyZQt2796NefPmIS4uDjExMYiKipJ++37zzTcRERGBrKwsAMD69evh6+uLMWPGGK3GiiQmJuLgwYPS4z179uDSpUtac8Wc668SE99KW2PXrl0Tc+fOFT169BB+fn6iR48eYubMmeLatWta/bKzs8Xjjz8ugoKChL+/v1i2bJnQaDQmqvpvGzdulO6vf/BfeeZavxBCHDhwQAwePFi4ubkJe3t7MWDAAPHpp59q9THn+oUQYtu2bdJ+79Wrlzh//rypS6pQYWGhGDx4sOjRo4cAIHr37i2mTJmi9bc+mzdvFj179hR9+/YVY8aMEdeuXRMZt++JozEJwv/Z5Tr/RqJx6D/EmYspdVq7SqUSVlZWOuf75s2bK6z/+vXrdVqXodavXy/69u0rBgwYIPr27Sv69esnvv/+eyGE0Lv/zaX+qlAIYUafYVAD9eHLvS19DJZcf2pqKjw8PAAAFy5cQNeuXU1cUdVVdf9Pf3cb/n1CflNB40Y2OLnhTfTy6lgXZVbIkucPYPn1P8j05/REZBJ733ke4sQ2PDqkl9bywuKSCi83qdT5KC7hlw41BAwHogauKiGRkn4Lk5duQUGR/D0Mql8YDkQEwLCQSLiSiqhf4zFm4Xrk5BWYqFIyBoYDEWnRFxIz/7kDAHDi7CWEvvER7qjUpiiRjIDhQEQ6VRQSZX6/cAW9X1yJjDv3jFwZGQPDgYj02vvO8yg5tgWebV1lbZdvZKL1lDfx4+nKP76DLAvDgYj0yisoxPi3NyI5NUtnuxDAyAUb4DB6Ls5fSTVydVRXzPazlYjIPNgpGyNqxVxcvpGJ81du4NyVGzh/JQ3HYhNxT/33m9LqgiJ0f/pdONkpcWrjfPh2bKu1ntJSDayt+fuopWA4EFGlrK2t4NWuJbzatcTkwYHS8vzCIoT98zN8/UuctEyVV6AzJF7esBdvzxyD1i2aGrt8qgbGOBFVW5PGtvhq+WyIE9swe8IglP/6obKQcB77Ms5fSUXC1TQMePkDpKTpvjxF5oXhQES1YvOrM6HRExK/xCfjSvotDJwXjsSr978xbfnOKNMUS5ViOJBJJScno1+/fggJCdHZfu/ePYSFhSE4OBgBAQFYvny57DsKyLxUFBJlxy3t1l30n7sasZf+wvJdUXhy1Q7TFEp6MRzIZHbv3o0nnnhC78d2h4WFwcbGBqdPn8bJkycRGRmJdevWGa9IqraykJg1sq+s7a46H0GzV0AIIOLIb3hk8SYTVEj6MBzIZFxcXPDzzz+jc+fOOtvPnTuHqKgozJ8/H8D9r/R88cUXsWrVKmg0GmOWStX06/k/cTxO9/dWlz//O3Dqfxi9aBOPqxnh3UpkMpV9+cmxY8fg4OCAbt26ScuCgoKQmZmJ+Ph42dep6pOaqv/++7JvD6Ta1c69Oba8+rh0++u5lBs4fyUVGh1XBk8lpODZrNs4GxgobySjs5hwyMnJ0duuVqtRWloKtdpyP+vF0sdQ3fqLi4tRWloqO8YXL16Eq6ur1nIHBwcAQEJCAh566CGDt1H2XQ2GyMvLq3S+mSNznD/OShsM8u2IQb5/fzfEpes3MXLhRtzJyZP1v5J5D60ffQt/bH0LLZs7GbPUGqvK/reE73uwmHCIjY3V267RaJCfn4/ExESz+OrJ6rD0MVS3/tu3b0tflFLe9evXodFotJZnZ2cDuP91jW3atKmdwh+QnJwMlUpVJ+uuS5YwfzQaDZZGnkRxSQkUivt/Xf2gvMJidHlyOfp6tsbCR/rAyU5p/EKroSr7v6IbMMyJxYRDYCWnmmq1GgkJCfD29oa9vb2Rqqpdlj4GtVqN+fPnY/fu3Xr7RUdHa31Ru4uLC9RqtewYe3h44OLFi1rLyy4PeXt7Vzonyrtw4YLe9oyMDAwZMgQA4OnpCU9PT4PXbS4sZf4c+v/vdgfun0Ucjb2I/164gjMX/0JGtkp6L+K35DRMXPMNxvT2wca5j6K5k51pCjaQpex/Q1lMOBhyGmZtbQ17e3uLOGWriKWP4bHHHsM777wjXf7RpUWLFrCx+XvqNWrUCNbW1rIxd+3aFV988YXW8tzcXACAj49PlfZRZV/7Wb5eOzs7i93/ljZ/enk7opf3/RsSys4eb5Y0xrxPvsbNbBU0QuD7/57Hod8T8HB/f3z6ZhiaO1U8t0zN0va/PuZ57kkWq0mTJnB3d0fLli0r/Fc+GPQZNmwYcnNzcfHi33e7xMTEwM3NDX5+fnU1BDKxMb19kbF/DaL+ORft3ZsDADRC4JuTZ+H6yOuYtGQz7qhyTVxl/cdwILPl5+eH8ePHIzw8HACQn5+PzZs3Y8GCBWZ7TZ1qz7i+fri6dxVDwkT4CiOT+e677xASEoLDhw/jjz/+QEhICLZv367VZ9euXSgsLERwcDD69euHyZMn49VXXzVRxWQKDAnTsJj3HKj+mTBhAiZMmKC3T9OmTbFnzx4jVUTmbFxfP4zr64fvf4vH3I+/wF8370ghceDUHzrfk/jgy8OY/9goE1ZtuXjmQEQWpSpnEm9v/xbTl28zZbkWi+FARBbJkJDQCIF/R8dg1Px1pi3WAjEciMii6QsJzf9/TseRM4no8+JKfnZTFTAciKhe0BUS5f1+4Qr8nnkXJSUlJqjO8jAciKhe6dOtI2wb6b7XJuFqGjzDliCvoNDIVVkehgMR1Ssvb9iLzOwcrS8aKu9qxm089PjbuJsr/+A/+htvZSWieuWLJc8BuP9BeH9cvo4jZxLwa0IKLl5LR9qte8grLELGHRU6zXgLiTuXo2VzZxNXbJ4YDkRUL1lZWSGgS3sEdGmvtbyoqAQ/xyfhWOwF7Dz8KxbOGG2iCs0bw4GIGhRbWxuE9vJGaC9vU5di1vieAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgMREckwHIiISIbhQEREMgwHIiKSYTgQEZEMw4GIiGQYDkREJMNwICIiGYYDERHJMByIiEiG4UBERDIMByIikmE4EBGRDMOBiIhkGA5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAcyiTt37mDZsmUYMGAAQkJC4O/vj/fffx8lJSVa/e7du4ewsDAEBwcjICAAy5cvhxDCRFUTNRw2pi6AGqZDhw4hMjISv/76K5ydnZGWloaAgAAUFRXh3XfflfqFhYXBxcUFp0+fRl5eHoKDg+Hk5IRXX33VhNUT1X88cyCTcHFxweuvvw5nZ2cAQOvWrTFlyhTs3btX6nPu3DlERUVh/vz5AAA7Ozu8+OKLWLVqFTQajUnqJmooeOZAJjF69GjZMqVSiaKiIunxsWPH4ODggG7duknLgoKCkJmZifj4ePj7+xu8vdTUVL3t6enpBq+LqCGwmHDIycnR265Wq1FaWgq1Wm2kimqfpY+hpvWfPHkSDz/8sHSsL168CFdXV61j7+DgAABISEjAQw89ZPC6PTw8DO6bl5dX6XwzRw19/phaVep3dHQ0QkU1YzHhEBsbq7ddo9EgPz8fiYmJsLKyzKtllj6GmtQfFxeHlJQULF68WDrW169fh0aj0Tr22dnZAIDExES0adOm9oovJzk5GSqVqk7WXZca8vwxB1WpPyQkxDhF1YDFhENgYKDedrVajYSEBHh7e8Pe3t5IVdUuSx+DWq3G/PnzsXv3br39oqOjERAQID1OS0vDJ598gq+//lrrUpGHhwcuXryodezLLg95e3tXOifKu3Dhgt72jIwMDBkyBADg6ekJT09Pg9dtLurD/GH95sNiwsGQ0zBra2vY29tbxClbRSx9DI899hjeeecd6fKPLi1atICNzf2pd+fOHcyYMQNbtmzBwIEDtfp17doVX3zxhda+yM3NBQD4+PhUaR917dpVb3v5eu3s7Cx2/1v6/GH95sPyzt3IrDVp0gTu7u5o2bJlhf/KgiEnJwfjx4/HO++8g+HDhwMAtm3bJq1r2LBhyM3NxcWLF6VlMTExcHNzg5+fn3EHRtTAMBzIJAoKCjBhwgT06dMHbdq0QUxMDGJiYrB161apj5+fH8aPH4/w8HAAQH5+PjZv3owFCxZY5DVpIktiMZeVqH7Zvn07oqOjER0djbVr11bYb9euXZg7dy6Cg4NRXFyMyZMn8w/giIyA4UAmMWfOHMyZM6fSfk2bNsWePXuMUBERlcdzcyIikmE4EBGRDMOBiIhkGA5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgMREckwHIiISIbhQEREMgwHIiKSYTgQEZEMw4GIiGQYDkREJMNwICIiGYYDERHJMByIiEiG4UBERDIMByIikmE4EBGRDMOBiIhkGA5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgOZRGFhIZYuXYrBgwdj+PDh6NmzJyZOnIiUlBStfvfu3UNYWBiCg4MREBCA5cuXQwhhoqqJGg4bUxdADVN2djb+9a9/4ezZs3B3d4dGo8H06dPx6KOP4syZM1K/sLAwuLi44PTp08jLy0NwcDCcnJzw6quvmrB6ovqP4UAm0bx5cxw8eBDu7u4AACsrKwwcOBBHjhyR+pw7dw5RUVFITEwEANjZ2eHFF1/E8uXLMW/ePFhZGX7im5qaqrc9PT29GqMgqr8sJhxycnL0tqvVapSWlkKtVhupotpn6WOoav2dO3eWjmtaWhp27NiB2bNnS8u+//57ODg4oG3bttIyb29vZGZm4rfffoOfn5/BtXl4eBjcNy8vr9L5Zo4a2vwxN1Wp39HR0QgV1YzFhENsbKzedo1Gg/z8fCQmJlbpN0pzYuljqE79WVlZePvtt3H16lVMmzYNo0ePlo71mTNn4OTkpHXsMzMzAQDHjx9HcXFx7Q8CQHJyMlQqVZ2suy41xPljTqpSf0hIiHGKqgGLCYfAwEC97Wq1GgkJCfD29oa9vb2Rqqpdlj6G6tY/atQopKenY/r06di5cyc2bNgAAHBwcICTk5PWsc/KygIAtGrVqtI5Ud6FCxf0tmdkZGDIkCEAAE9PT3h6ehq8bnPRUOePubD0+h9kMeFgyGmYtbU17O3tLeKUrSKWPobdu3cjIiJCb58zZ86gV69eWsscHR3xwQcfIDQ0FG+++SZ8fHzg7OyM4uJirX1x9+5dAICLi0uV9lHXrl31tjs4OEg/29nZWez+t/T5w/rNh8WEA1mGRx99FIsXL9b6z/ZBLVq0QGlpKYD7L6YyXl5eAIDExET4+PigU6dOuHnzptZzMzIyAACdOnWq7dKJqBzLu7BHZq1JkyZwd3dHy5YtK/xnY2OD3bt346OPPtJ6btkdQ61btwYADBs2DLm5ubh48aLUJyYmBm5ublV6M5qIqo7hQCbz2Wef4datWwCAgoICvPfee/D19UVQUBAAwM/PD+PHj0d4eDgAID8/H5s3b8aCBQss8g1LIkvCy0pkEsOGDUNsbCxGjBgBBwcH5ObmwsfHB4cOHYKtra3Ub9euXZg7dy6Cg4NRXFyMyZMn8w/giIyA4UAm4eHhId2VpE/Tpk2xZ88eI1REROXx3JyIiGQYDkREJKMQ/IhLIqSmpkofsXH9+nW0bdvWxBURmRbDgQhASUmJ9DcUZbfbEjVkDAciIpLhew5ERCTDcCAiIhmGAxERyTAciIhIhuFAREQyDAciIpJhOBARkQzDgYiIZBgOREQkw3AgIiIZhgMREckwHIiISIbhQEREMgwHIiKSYTgQEZEMw4GIiGQYDkREJPN/v/9tPRePhQQAAAAASUVORK5CYII=", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "d = numpy.array((-1,0.5))\n", "vectors = []\n", "for _ in range(30):\n", " m = randint(-10,10)\n", " n = randint(-10,10)\n", " vectors.append(m*a + n*d)\n", " \n", "plot_vector(vectors)\n", "pyplot.title(\"Thirty linear combinations of the vectors $\\\\mathbf{a}$ and $\\\\mathbf{d}$.\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*What's going on?*\n", "\n", "Well, the new vector $\\mathbf{d}$ happens to be a scaled version of the original vector $\\mathbf{a}$—we say that they are _colinear_. Thus, all linear combinations of $\\mathbf{a}$ and $\\mathbf{d}$ end up on one line, which is their span. Their combinations are not able to travel all over the plane!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Definition:\n", "\n", "> A **basis** for a vector space is a set of _linearly independent_ vectors that _span_ that space.\n", "\n", "We saw above that $\\mathbf{d}$ is a scalar multiplied by $\\mathbf{a}$: it is linearly _dependent_ with $\\mathbf{a}$. The vector $\\mathbf{b}$, however, is linearly independent with $\\mathbf{a}$. Bring in vector $\\mathbf{c}$ now: it can be written as a linear combination of $\\mathbf{a}$ and $\\mathbf{b}$: $\\alpha\\, \\mathbf{a} + \\beta\\, \\mathbf{b} = \\mathbf{c}$, for some scalars $\\alpha$ and $\\beta$. In 2D space, any third vector will be linearly dependent with $\\mathbf{a}$ and $\\mathbf{b}$: these two form a _full set_ of independent vectors (and a basis).\n", "\n", "In 3D space, two vectors that are linearly independent span a plane. We need a third vector that is not a linear combination of the first two to span the whole space, and form a basis. Any fourth vector will be linearly dependent as it can be written as a linear combination of the basis vectors.\n", "\n", "##### Key idea:\n", "\n", "> In a set of linearly independent vectors, no one vector can be written as a linear combination of the others. The only way to get the zero vector from a linear combination of all the vectors is to multiply them all by zero." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Matrices" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### What's a matrix?\n", "\n", "In many books, they'll tell you that a matrix is a \"table\" of numbers, ordered in rows and columns. Maybe that's enough for some people, but you will get a kick out of _seeing_ what a matrix does!\n", "\n", "Let's remember our friendly vectors from above:\n", "\n", "$$\n", " \\mathbf{a} = \\left[ \\begin{array}{c} -2 \\\\ 1 \\end{array} \\right], \\quad \n", " \\mathbf{b} = \\left[ \\begin{array}{c} 1 \\\\ -3 \\end{array} \\right] \n", "$$\n", "\n", "Our little experiment with 30 random linear combinations of $\\mathbf{a}$ and $\\mathbf{b}$ helped us visualize that they can span the 2D space, and nothing is stopping us from using them as a basis if we so desire.\n", "\n", "Remember also our vector $\\mathbf{c} = \\left[ \\begin{array}{c} 2 \\\\ 1 \\end{array} \\right]$. Choosing $\\mathbf{i}$ and $\\mathbf{j}$ as a basis, then $\\mathbf{c} = 2\\,\\mathbf{i} + 1\\,\\mathbf{j}$.\n", "\n", "Now imagine that we use the components of $\\mathbf{c}$ to make a linear combination of $\\mathbf{a}$ and $\\mathbf{b}$:\n", "\n", "$$\n", " 2\\,\\mathbf{a} + 1\\,\\mathbf{b} =\n", " 2\\cdot\\left[ \\begin{array}{c} -2 \\\\ 1 \\end{array} \\right] +\n", " 1\\cdot\\left[ \\begin{array}{c} 1 \\\\ -3 \\end{array} \\right] = \n", " \\left[ \\begin{array}{c} -3 \\\\ -1 \\end{array} \\right]\n", "$$\n", "\n", "This is a new vector, let's call it $\\mathbf{c}^\\prime$: \n", "\n", "- it has components $\\left[ \\begin{array}{c} 2 \\\\ 1 \\end{array} \\right]$ in the $\\mathbf{a}$, $\\mathbf{b}$ system of coordinates, and \n", "- it has components\n", "$\\left[ \\begin{array}{c} -3 \\\\ -1 \\end{array} \\right]$ in the $\\mathbf{i}$, $\\mathbf{j}$ system of coordinates.\n", "\n", "This will blow your mind. Arrange the vectors $\\mathbf{a}$ and $\\mathbf{b}$ as the columns of a matrix, and you'll see that:\n", "\n", "$$\n", " \\begin{bmatrix} -2 & 1 \\\\ \n", " 1 & -3 \\end{bmatrix} \n", " \\left[ \\begin{array}{c} 2 \\\\ 1 \\end{array} \\right] =\n", " \\left[ \\begin{array}{c} -3 \\\\ -1 \\end{array} \\right]\n", "$$\n", "\n", "The matrix $\\,A=\\begin{bmatrix} -2 & 1 \\\\ \n", " 1 & -3 \\end{bmatrix} $ when multiplied by the vector $\\mathbf{c}$ gives the vector $\\mathbf{c}^\\prime$.\n", "\n", "##### Key idea:\n", "\n", "> The matrix $A$ represents the **linear transformation** that takes vector $\\mathbf{c}$ and transforms it into $\\mathbf{c}^\\prime$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Matrices with NumPy\n", "\n", "Let's play around with this a bit more.\n", "\n", "We can define a NumPy array to represent the matrix $A$, as follows:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[-2 1]\n", " [ 1 -3]]\n" ] } ], "source": [ "A = [[-2,1], [1,-3]]\n", "A = numpy.array(A)\n", "print(A)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can use the NumPy [`dot()`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.dot.html) method to multiply the matrix $A$ and the vector $\\mathbf{c}$:" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([-3, -1])" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.dot(c)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Sure enough, this gives the vector $\\mathbf{c}^\\prime$. Now let's see what happens when we multiply the matrix $A$ with the basis vectors $\\mathbf{i}$ and $\\mathbf{j}$:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([-2, 1])" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.dot(i)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 1, -3])" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.dot(j)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Woot! The matrix $A$ when multiplied by the vector $\\mathbf{i}$ gives the vector $\\mathbf{a}$ and when multiplied by the vector $\\mathbf{j}$ gives the vector $\\mathbf{b}$.\n", "\n", "Remember that we used the components of $\\mathbf{c}$ in a linear combination of $\\mathbf{a}$ and $\\mathbf{b}$ to get $\\mathbf{c}^\\prime$: $2\\,\\mathbf{a} + 1\\,\\mathbf{b}$.\n", "What we find is that the linear transformation represented by the matrix $A$, transforms the vector $\\mathbf{c}$ to the linear combination of the transformed $\\mathbf{i}$ and $\\mathbf{j}$.\n", "\n", "Since _all_ vectors are a linear combination of the basis vectors, $\\mathbf{i}$ and $\\mathbf{j}$, scaled by the vector components, they are _all_ transformed to a linear combination of $\\mathbf{a}$ and $\\mathbf{b}$ with the same scalars." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's try to visualize that. Our helpful custom function `plot_linear_transformation()` draws a grid of points on the plane, then applies the linear transformation described by the matrix argument, and plots the transformed grid. This is what $A$ does to the grid:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot_linear_transformation(A)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Key idea:\n", "\n", "> A **linear transformation** keeps the origin in place and transforms straight lines to straight lines.\n", "\n", "The third episode of the series [_\"Essence of Linear Algebra\"_](http://3b1b.co/eola) uses wonderful animations to illustrate the idea of matrices as linear transformations [3]." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's practice with another example. Consider the matrix:\n", "\n", "$$ M = \\begin{bmatrix} 1 & 2 \\\\\n", " 2 & 1 \\end{bmatrix} $$\n", "\n", "The first column corresponds to the vector where $\\mathbf{i}$ lands after the transformation, and the second column is where $\\mathbf{j}$ lands:\n", "$$\n", "\\mathbf{i} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix} \\\\\n", "\\mathbf{j} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 2 \\\\ 1 \\end{bmatrix}\n", "$$\n", "\n", "Any arbitrary vector on the plane, $\\mathbf{x} = \\left[ \\begin{array}{c} x \\\\ y \\end{array} \\right]$, is transformed to: \n", "\n", "$$\n", " x \\left[ \\begin{array}{c} 1 \\\\ 2 \\end{array} \\right] + \n", " y \\left[ \\begin{array}{c} 2 \\\\ 1 \\end{array} \\right]\n", "$$" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1 2]\n", " [2 1]]\n" ] } ], "source": [ "M = [[1,2], [2,1]]\n", "M = numpy.array(M)\n", "print(M)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([1, 2])" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.dot(i)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([2, 1])" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.dot(j)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_linear_transformation(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To see what the transformation does to one particular vector, we can use our custom function `plot_vector` again, using the vector and its transformed self. Try several different ones…" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = numpy.array((0.5,1))\n", "\n", "vectors = [x, M.dot(x)]\n", "plot_vector(vectors)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Note:\n", "\n", "When we represent a matrix in Python using the NumPy array data type, we define it by listing the **rows** of the matrix. We didn't emphasize this in the examples above, where we chose matrices whose rows matched the columns! But that's just a coincidence. Look at another case:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 1 2]\n", " [-3 2]]\n" ] } ], "source": [ "N = numpy.array([[1,2],[-3,2]])\n", "print(N)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_linear_transformation(N)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Exercise:\n", "\n", "Create a $2\\times2$ matrix of your choosing (as a NumPy array of the row list), print it, then multiply it by the basis vectors $\\mathbf{i}$ and $\\mathbf{j}$, and finally visualize it using our helper function `plot_linear_transformation()`." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Matrix-vector multiplication\n", "\n", "Consider again the matrix $A$ defined above. As a linear transformation, it transforms any arbitrary vector on the plane, $\\mathbf{x} = \\left[ \\begin{array}{c} x \\\\ y \\end{array} \\right]$ , to: \n", "$$\n", " x \\left[ \\begin{array}{c} -2 \\\\ 1 \\end{array} \\right] + \n", " y \\left[ \\begin{array}{c} 1 \\\\ -3 \\end{array} \\right]\n", "$$\n", "\n", "Since applying the linear transformation *is* computing the matrix-vector multiplication $A\\mathbf{x}$, we see that matrix-vector multiplication is a combination of the matrix columns scaled by the vector components:\n", "\n", "$$\n", " A\\mathbf{x} = x\\,\\mathbf{a} + y\\,\\mathbf{b}\n", "$$\n", "where the vectors $\\mathbf{a}$ and $\\mathbf{b}$ are the columns of $A$.\n", "\n", "##### Key idea:\n", "\n", "> The matrix-vector multiplication $A\\mathbf{x}$ is a linear combination of the columns of $A$ scaled by the components of $\\mathbf{x}$.\n", "\n", "This is also the case in 3 dimensions. Consider the matrix $B$ and the vector $\\mathbf{y}=\\left[ \\begin{array}{c} x \\\\ y \\\\z \\end{array} \\right]$\n", "\n", "$$ B = \\begin{bmatrix} 1 & 2 & 4\\\\\n", " 2 & 1 & -1\\\\\n", " 0 & 3 & 1 \\end{bmatrix} $$\n", "\n", "$$ B\\mathbf{y} = x\\left[ \\begin{array}{c} 1 \\\\ 2 \\\\0 \\end{array} \\right] + \n", " y\\left[ \\begin{array}{c} 2 \\\\ 1 \\\\3 \\end{array} \\right] + \n", " z\\left[ \\begin{array}{c} 4 \\\\ -1 \\\\1 \\end{array} \\right] \n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Rotation\n", "\n", "Imagine that you want a transformation that takes any vector and rotates it 90 degrees to the left. You can visualize that the unit vectors need to be transformed as follows:\n", "\n", "$$\n", "\\mathbf{i} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\\\\n", "\\mathbf{j} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} -1 \\\\ 0 \\end{bmatrix}\n", "$$\n", "\n", "That means that the matrix that transforms all vectors by a left 90-degree **rotation** is:\n", "\n", "$$ R = \\begin{bmatrix} 0 & -1 \\\\\n", " 1 & 0 \\end{bmatrix} $$\n", "\n", "The rotation of any vector $\\mathbf{x}$ is the multiplication $R\\mathbf{x}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Shear\n", "\n", "Another special transformation turns every square into a diamond shape by leaving $\\mathbf{x}$ unchanged, and transforming $\\,\\mathbf{j}$ so its tip falls on the coordinates $(1,1)$: \n", "\n", "$$\n", "\\mathbf{i} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\\\\n", "\\mathbf{j} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}\n", "$$\n", "\n", "This transformation is often called **shear**, and the matrix is:\n", "\n", "\n", "$$ S = \\begin{bmatrix} 1 & 1 \\\\\n", " 0 & 1 \\end{bmatrix} $$\n", "\n", "The shear of any vector $\\mathbf{x}$ is the multiplication $S\\mathbf{x}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's define these matrices as NumPy arrays, then use our helper function to visualize the corresponding transformation." ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0 -1]\n", " [ 1 0]]\n" ] } ], "source": [ "rotation = numpy.array([[0,-1], [1,0]])\n", "print(rotation)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1 1]\n", " [0 1]]\n" ] } ], "source": [ "shear = numpy.array([[1,1], [0,1]])\n", "print(shear)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot_linear_transformation(shear)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Scaling\n", "\n", "A **scaling** transformation doesn't rotate or shear the basis vectors, but scales them in length. For example, a transformation that elongates $\\mathbf{i}$ but shrinks $\\mathbf{j}$ could do:\n", "\n", "\n", "$$\n", "\\mathbf{i} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 2 \\\\ 0 \\end{bmatrix} \\\\\n", "\\mathbf{j} = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\Rightarrow \\begin{bmatrix} 0 \\\\ 0.5 \\end{bmatrix}\n", "$$\n", "\n", "Look at what the matrix transformation does in this case…" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[2. 0. ]\n", " [0. 0.5]]\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "scale = numpy.array([[2,0], [0,0.5]])\n", "print(scale)\n", "plot_linear_transformation(scale)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Identity\n", "\n", "The common structure of scaling matrices is that they have non-zero values in the diagonal, but zero values elsewhere.\n", "One special scaling matrix leaves the lengths of the basis vectors unchanged: it is called the **identity** matrix:\n", "\n", "\n", "$$ I = \\begin{bmatrix} 1 & 0 \\\\\n", " 0 & 1 \\end{bmatrix} $$\n", "\n", "NumPy creates identity arrays of any size with [`numpy.identity()`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.eye.html?highlight=eye#numpy.eye), passing the dimension (number of rows and columns) as argument. In the 2D case:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1. 0.]\n", " [0. 1.]]\n" ] } ], "source": [ "I = numpy.identity(2)\n", "print(I)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Exercise:\n", "\n", "Create a rotation matrix that rotates every vector 90 degrees clockwise, then visualize the transformation with our helper function." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What do you think happens when we apply two linear transformations? For example, if we rotate all vectors by 90 degrees to the left, and *then* apply a shear transformation?\n", "\n", "Take any vector $\\mathbf{x}$, rotate it by multiplying it with the matrix $R$, then take this transformed vector and multiply it by the matrix $S$. The final vector is the result of the two combined linear transformations. It is analogous to the composition of two functions, and its computation leads to a matrix-matrix multiplication:\n", "\n", "$$\n", " S\\, R\\, \\mathbf{x} =\n", " \\begin{bmatrix} 1 & 1 \\\\\n", " 0 & 1 \\end{bmatrix}\n", " \\begin{bmatrix} 0 & -1 \\\\\n", " 1 & 0 \\end{bmatrix}\n", " \\mathbf{x}\n", "$$\n", "\n", "We can almost work this out in our heads. \n", "The unit vector $\\mathbf{i}$ gets first rotated to $\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$ (the first column of $R$), and then is transformed by $S$ via the multiplication:\n", "\n", "$$\n", " S\\,\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} =\n", " \\begin{bmatrix} 1 & 1 \\\\\n", " 0 & 1 \\end{bmatrix}\n", " \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} =\n", " 0 \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}+\n", " 1 \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} =\n", " \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}\n", "$$\n", "\n", "Similarly for the unit vector $\\mathbf{j}$, we now find it lands on $\\begin{bmatrix} -1 \\\\ 0 \\end{bmatrix}$. We have the two columns of the resulting matrix from the multiplication $SR$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Matrix-matrix multiplication" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Python multiplies matrices\n", "\n", "Python (since version 3.5) has a **built-in operator** that computes matrix-matrix multiplication: `@`. Try it:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 1 -1]\n", " [ 1 0]]\n" ] } ], "source": [ "print(shear@rotation)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Yep. Those are the two column vectors we worked out above. Beautiful. \n", "Let's visualize the combined transformation now." ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot_linear_transformation(shear@rotation)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Key idea:\n", "\n", "> Matrix multiplication corresponds to composition of linear transformations, i.e., applying two transformations in sequence.\n", "\n", "This view of matrix multiplication will save you from a lot of unnecessary memorization. It also illuminates the properties of matrix multiplications. For example, is it the same to apply shear and *then* rotate, instead of the other way around? \n", "\n", "This is the same question as asking if matrix multiplication is commutative. Is $SR$ the same as $RS$?\n", "\n", "We have a helper function that plots the grid lines on a plane after two transformations in sequence. Let's try it with $S$ and $R$ in swapped orders." ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# the order of transformation: from right to left\n", "plot_linear_transformations(rotation, shear) " ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot_linear_transformations(shear, rotation) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nope. The result is not the same. The order of transformations matters and **matrix mulitiplication is not commutative** (in general).\n", "\n", "Episode four of the series [_\"Essence of Linear Algebra\"_](http://3b1b.co/eola) beautifully illustrates the key idea of matrix multiplication as composition of linear transformations [4]." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Inverse of a matrix" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thinking of matrices as linear transformations also helps demistify the idea of an **inverse**. We won't go into details here, but imagine that you apply two transformations in sequence, and every vector in 2D space ends up just where it started. Well, that can happen when one transformation undoes the previous one. This means that the second transformation is the inverse of the first.\n", "\n", "NumPy has great built-in Linear Algebra capabilities in the `numpy.linalg` module. Among its many functions, we get [`inv()`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.linalg.inv.html), to compute the inverse of a matrix. So we can try right away to visualize a sequence of transformations: first with the matrix $M$, then the inverse of $M$. Check it out." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [], "source": [ "from numpy.linalg import inv" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "M = numpy.array([[1,2], [2,1]])\n", "M_inv = inv(M)\n", "plot_linear_transformations(M, M_inv)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The combined action of the linear transformation $M$ and its inverse $M^{-1}$ is to leave every vector the same. In other words, the matrix multiplication $M^{-1}M$ is equal to the identity $I$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## What we've learned\n", "\n", "- What is a vector.\n", "- The two fundamental vector operations: vector addition and scaling.\n", "- The concept of basis vectors.\n", "- Making a linear combination of vectors; the concept of span.\n", "- When a set of vectors is linearly independent.\n", "- A matrix is a linear transformation.\n", "- Matrix-vector multiplication: a combination of the matrix columns scaled by the vector components.\n", "- Some special transformations: rotation, shear, scaling, identity.\n", "- Matrix-matrix multiplication: composition of linear transformations.\n", "- Idea of the matrix inverse: the transformation that reverses the effect of another." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\n", "\n", "1. Vectors, what even are they? Essence of linear algebra, chapter 1. Video at https://youtu.be/fNk_zzaMoSs (2016), by Grant Sanderson.\n", "2. Linear combinations, span, and basis vectors. Essence of linear algebra, chapter 2. Video at https://youtu.be/k7RM-ot2NWY (2016), by Grant Sanderson.\n", "3. Linear transformations and matrices. Essence of linear algebra, chapter 3. Video at https://youtu.be/kYB8IZa5AuE (2016), by Grant Sanderson.\n", "4. Matrix multiplication as composition. Essence of linear algebra, chapter 4. Video at https://youtu.be/XkY2DOUCWMU (2016), by Grant Sanderson." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.4" } }, "nbformat": 4, "nbformat_minor": 2 }