diff --git a/math_linear_algebra.ipynb b/math_linear_algebra.ipynb index 4233cb7..26ed9c5 100644 --- a/math_linear_algebra.ipynb +++ b/math_linear_algebra.ipynb @@ -17,10 +17,10 @@ "source": [ "\n", " \n", " \n", "
\n", - " \"Open\n", + " \"Open\n", " \n", - " \n", + " \n", "
" ] @@ -77,7 +77,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ @@ -93,11 +93,12 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", + "\n", "video = np.array([10.5, 5.2, 3.25, 7.0])\n", "video" ] @@ -111,7 +112,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ @@ -129,7 +130,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 4, "metadata": {}, "outputs": [], "source": [ @@ -146,11 +147,10 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ - "%matplotlib inline\n", "import matplotlib.pyplot as plt" ] }, @@ -164,13 +164,8 @@ }, { "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": true, - "jupyter": { - "outputs_hidden": true - } - }, + "execution_count": 6, + "metadata": {}, "outputs": [], "source": [ "u = np.array([2, 5])\n", @@ -186,7 +181,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 7, "metadata": {}, "outputs": [], "source": [ @@ -206,7 +201,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 8, "metadata": {}, "outputs": [], "source": [ @@ -225,7 +220,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 9, "metadata": {}, "outputs": [], "source": [ @@ -246,7 +241,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 10, "metadata": {}, "outputs": [], "source": [ @@ -263,7 +258,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 11, "metadata": {}, "outputs": [], "source": [ @@ -285,7 +280,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 12, "metadata": {}, "outputs": [], "source": [ @@ -311,12 +306,12 @@ "\n", "$\\left \\Vert \\textbf{u} \\right \\| = \\sqrt{\\sum_{i}{\\textbf{u}_i}^2}$\n", "\n", - "We could implement this easily in pure python, recalling that $\\sqrt x = x^{\\frac{1}{2}}$" + "That's the square root of the sum of all the squares of the components of $\\textbf{u}$. We could implement this easily in pure python, recalling that $\\sqrt x = x^{\\frac{1}{2}}$" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 13, "metadata": {}, "outputs": [], "source": [ @@ -337,11 +332,12 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "import numpy.linalg as LA\n", + "\n", "LA.norm(u)" ] }, @@ -354,7 +350,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 15, "metadata": {}, "outputs": [], "source": [ @@ -362,6 +358,7 @@ "plt.gca().add_artist(plt.Circle((0,0), radius, color=\"#DDDDDD\"))\n", "plot_vector2d(u, color=\"red\")\n", "plt.axis([0, 8.7, 0, 6])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -383,7 +380,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 16, "metadata": {}, "outputs": [], "source": [ @@ -402,7 +399,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 17, "metadata": { "scrolled": true }, @@ -414,6 +411,7 @@ "plot_vector2d(u, origin=v, color=\"r\", linestyle=\"dotted\")\n", "plot_vector2d(u+v, color=\"g\")\n", "plt.axis([0, 9, 0, 7])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.text(0.7, 3, \"u\", color=\"r\", fontsize=18)\n", "plt.text(4, 3, \"u\", color=\"r\", fontsize=18)\n", "plt.text(1.8, 0.2, \"v\", color=\"b\", fontsize=18)\n", @@ -441,7 +439,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 18, "metadata": {}, "outputs": [], "source": [ @@ -468,6 +466,7 @@ "plt.text(3.5, 0.4, \"v\", color=\"r\", fontsize=18)\n", "\n", "plt.axis([0, 6, 0, 5])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -489,7 +488,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 19, "metadata": {}, "outputs": [], "source": [ @@ -507,7 +506,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 20, "metadata": {}, "outputs": [], "source": [ @@ -530,6 +529,7 @@ "plot_vector2d(k * t3, color=\"b\", linestyle=\":\")\n", "\n", "plt.axis([0, 9, 0, 9])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -567,17 +567,18 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 21, "metadata": {}, "outputs": [], "source": [ - "plt.gca().add_artist(plt.Circle((0,0),1,color='c'))\n", + "plt.gca().add_artist(plt.Circle((0, 0), 1, color='c'))\n", "plt.plot(0, 0, \"ko\")\n", - "plot_vector2d(v / LA.norm(v), color=\"k\")\n", - "plot_vector2d(v, color=\"b\", linestyle=\":\")\n", - "plt.text(0.3, 0.3, \"$\\hat{u}$\", color=\"k\", fontsize=18)\n", + "plot_vector2d(v / LA.norm(v), color=\"k\", zorder=10)\n", + "plot_vector2d(v, color=\"b\", linestyle=\":\", zorder=15)\n", + "plt.text(0.3, 0.3, r\"$\\hat{u}$\", color=\"k\", fontsize=18)\n", "plt.text(1.5, 0.7, \"$u$\", color=\"b\", fontsize=18)\n", "plt.axis([-1.5, 5.5, -1.5, 3.5])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -604,7 +605,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 22, "metadata": {}, "outputs": [], "source": [ @@ -618,16 +619,16 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "But a *much* more efficient implementation is provided by NumPy with the `dot` function:" + "But a *much* more efficient implementation is provided by NumPy with the `np.dot()` function:" ] }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 23, "metadata": {}, "outputs": [], "source": [ - "np.dot(u,v)" + "np.dot(u, v)" ] }, { @@ -639,7 +640,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 24, "metadata": {}, "outputs": [], "source": [ @@ -655,7 +656,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 25, "metadata": {}, "outputs": [], "source": [ @@ -694,13 +695,13 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 26, "metadata": {}, "outputs": [], "source": [ "def vector_angle(u, v):\n", " cos_theta = u.dot(v) / LA.norm(u) / LA.norm(v)\n", - " return np.arccos(np.clip(cos_theta, -1, 1))\n", + " return np.arccos(cos_theta.clip(-1, 1))\n", "\n", "theta = vector_angle(u, v)\n", "print(\"Angle =\", theta, \"radians\")\n", @@ -730,7 +731,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 27, "metadata": {}, "outputs": [], "source": [ @@ -750,6 +751,7 @@ "plt.text(0.8, 3, \"$u$\", color=\"r\", fontsize=18)\n", "\n", "plt.axis([0, 8, 0, 5.5])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -778,7 +780,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 28, "metadata": {}, "outputs": [], "source": [ @@ -797,7 +799,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 29, "metadata": {}, "outputs": [], "source": [ @@ -829,7 +831,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 30, "metadata": {}, "outputs": [], "source": [ @@ -845,7 +847,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 31, "metadata": {}, "outputs": [], "source": [ @@ -872,7 +874,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 32, "metadata": {}, "outputs": [], "source": [ @@ -888,7 +890,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 33, "metadata": {}, "outputs": [], "source": [ @@ -904,7 +906,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 34, "metadata": {}, "outputs": [], "source": [ @@ -920,7 +922,7 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 35, "metadata": {}, "outputs": [], "source": [ @@ -929,7 +931,7 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 36, "metadata": {}, "outputs": [], "source": [ @@ -1000,7 +1002,7 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 37, "metadata": {}, "outputs": [], "source": [ @@ -1016,7 +1018,7 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 38, "metadata": {}, "outputs": [], "source": [ @@ -1045,7 +1047,7 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 39, "metadata": {}, "outputs": [], "source": [ @@ -1080,7 +1082,7 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 40, "metadata": {}, "outputs": [], "source": [ @@ -1090,7 +1092,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 41, "metadata": {}, "outputs": [], "source": [ @@ -1099,7 +1101,7 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 42, "metadata": {}, "outputs": [], "source": [ @@ -1115,7 +1117,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 43, "metadata": {}, "outputs": [], "source": [ @@ -1131,7 +1133,7 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 44, "metadata": {}, "outputs": [], "source": [ @@ -1142,7 +1144,7 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 45, "metadata": {}, "outputs": [], "source": [ @@ -1174,7 +1176,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 46, "metadata": {}, "outputs": [], "source": [ @@ -1190,7 +1192,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 47, "metadata": {}, "outputs": [], "source": [ @@ -1208,7 +1210,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 48, "metadata": {}, "outputs": [], "source": [ @@ -1217,7 +1219,7 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 49, "metadata": {}, "outputs": [], "source": [ @@ -1233,7 +1235,7 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 50, "metadata": {}, "outputs": [], "source": [ @@ -1242,7 +1244,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 51, "metadata": {}, "outputs": [], "source": [ @@ -1298,7 +1300,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Let's multiply two matrices in NumPy, using `ndarray`'s `dot` method:\n", + "Let's multiply two matrices in NumPy, using `ndarray`'s `np.matmul()` function:\n", "\n", "$E = AD = \\begin{bmatrix}\n", " 10 & 20 & 30 \\\\\n", @@ -1317,7 +1319,7 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 52, "metadata": {}, "outputs": [], "source": [ @@ -1326,10 +1328,42 @@ " [11, 13, 17, 19],\n", " [23, 29, 31, 37]\n", " ])\n", - "E = A.dot(D)\n", + "E = np.matmul(A, D)\n", "E" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Python 3.5 [introduced](https://docs.python.org/3/whatsnew/3.5.html#pep-465-a-dedicated-infix-operator-for-matrix-multiplication) the `@` infix operator for matrix multiplication, and NumPy 1.10 added support for it. `A @ D` is equivalent to `np.matmul(A, D)`:" + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "metadata": {}, + "outputs": [], + "source": [ + "A @ D" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The `@` operator also works for vectors: `u @ v` computes the dot product of `u` and `v`:" + ] + }, + { + "cell_type": "code", + "execution_count": 54, + "metadata": {}, + "outputs": [], + "source": [ + "u @ v" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -1339,7 +1373,7 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 55, "metadata": {}, "outputs": [], "source": [ @@ -1348,7 +1382,7 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 56, "metadata": {}, "outputs": [], "source": [ @@ -1366,12 +1400,12 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 57, "metadata": {}, "outputs": [], "source": [ "try:\n", - " D.dot(A)\n", + " D @ A\n", "except ValueError as e:\n", " print(\"ValueError:\", e)" ] @@ -1387,7 +1421,7 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 58, "metadata": {}, "outputs": [], "source": [ @@ -1396,16 +1430,16 @@ " [4,1],\n", " [9,3]\n", " ])\n", - "A.dot(F)" + "A @ F" ] }, { "cell_type": "code", - "execution_count": 58, + "execution_count": 59, "metadata": {}, "outputs": [], "source": [ - "F.dot(A)" + "F @ A" ] }, { @@ -1417,7 +1451,7 @@ }, { "cell_type": "code", - "execution_count": 59, + "execution_count": 60, "metadata": {}, "outputs": [], "source": [ @@ -1426,16 +1460,16 @@ " [2, 5, 1, 0, 5],\n", " [9, 11, 17, 21, 0],\n", " [0, 1, 0, 1, 2]])\n", - "A.dot(D).dot(G) # (AB)G" + "(A @ D) @ G # (AD)G" ] }, { "cell_type": "code", - "execution_count": 60, + "execution_count": 61, "metadata": {}, "outputs": [], "source": [ - "A.dot(D.dot(G)) # A(BG)" + "A @ (D @ G) # A(DG)" ] }, { @@ -1445,22 +1479,22 @@ "It is also ***distributive* over addition** of matrices, meaning that $(Q + R)S = QS + RS$. For example:" ] }, - { - "cell_type": "code", - "execution_count": 61, - "metadata": {}, - "outputs": [], - "source": [ - "(A + B).dot(D)" - ] - }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [], "source": [ - "A.dot(D) + B.dot(D)" + "(A + B) @ D" + ] + }, + { + "cell_type": "code", + "execution_count": 63, + "metadata": {}, + "outputs": [], + "source": [ + "A @ D + B @ D" ] }, { @@ -1478,22 +1512,22 @@ "For example:" ] }, - { - "cell_type": "code", - "execution_count": 63, - "metadata": {}, - "outputs": [], - "source": [ - "A.dot(np.eye(3))" - ] - }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [], "source": [ - "np.eye(2).dot(A)" + "A @ np.eye(3)" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "metadata": {}, + "outputs": [], + "source": [ + "np.eye(2) @ A" ] }, { @@ -1505,7 +1539,7 @@ }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 66, "metadata": { "scrolled": true }, @@ -1514,38 +1548,6 @@ "A * B # NOT a matrix multiplication" ] }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**The @ infix operator**\n", - "\n", - "Python 3.5 [introduced](https://docs.python.org/3/whatsnew/3.5.html#pep-465-a-dedicated-infix-operator-for-matrix-multiplication) the `@` infix operator for matrix multiplication, and NumPy 1.10 added support for it. If you are using Python 3.5+ and NumPy 1.10+, you can simply write `A @ D` instead of `A.dot(D)`, making your code much more readable (but less portable). This operator also works for vector dot products." - ] - }, - { - "cell_type": "code", - "execution_count": 66, - "metadata": {}, - "outputs": [], - "source": [ - "import sys\n", - "print(\"Python version: {}.{}.{}\".format(*sys.version_info))\n", - "print(\"Numpy version:\", np.version.version)\n", - "\n", - "# Uncomment the following line if your Python version is ≥3.5\n", - "# and your NumPy version is ≥1.10:\n", - "\n", - "#A @ D" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note: `Q @ R` is actually equivalent to `Q.__matmul__(R)` which is implemented by NumPy as `np.matmul(Q, R)`, not as `Q.dot(R)`. The main difference is that `matmul` does not support scalar multiplication, while `dot` does, so you can write `Q.dot(3)`, which is equivalent to `Q * 3`, but you cannot write `Q @ 3` ([more details](http://stackoverflow.com/a/34142617/38626))." - ] - }, { "cell_type": "markdown", "metadata": {}, @@ -1645,7 +1647,7 @@ "metadata": {}, "outputs": [], "source": [ - "(A.dot(D)).T" + "(A @ D).T" ] }, { @@ -1654,7 +1656,7 @@ "metadata": {}, "outputs": [], "source": [ - "D.T.dot(A.T)" + "D.T @ A.T" ] }, { @@ -1679,7 +1681,7 @@ "metadata": {}, "outputs": [], "source": [ - "D.dot(D.T)" + "D @ D.T" ] }, { @@ -1830,6 +1832,8 @@ "x_coords_P, y_coords_P = P\n", "plt.scatter(x_coords_P, y_coords_P)\n", "plt.axis([0, 5, 0, 4])\n", + "plt.gca().set_aspect(\"equal\")\n", + "plt.grid()\n", "plt.show()" ] }, @@ -1851,6 +1855,7 @@ "plt.plot(x_coords_P, y_coords_P, \"bo\")\n", "plt.plot(x_coords_P, y_coords_P, \"b--\")\n", "plt.axis([0, 5, 0, 4])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -1871,6 +1876,7 @@ "from matplotlib.patches import Polygon\n", "plt.gca().add_artist(Polygon(P.T))\n", "plt.axis([0, 5, 0, 4])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -1915,9 +1921,10 @@ "plt.text(2.5, 0.5, \"$H_{*,1}$\", color=\"k\", fontsize=18)\n", "plt.text(4.1, 3.5, \"$H_{*,2}$\", color=\"k\", fontsize=18)\n", "plt.text(0.4, 2.6, \"$H_{*,3}$\", color=\"k\", fontsize=18)\n", - "plt.text(4.4, 0.2, \"$H_{*,4}$\", color=\"k\", fontsize=18)\n", + "plt.text(4.3, 0.2, \"$H_{*,4}$\", color=\"k\", fontsize=18)\n", "\n", "plt.axis([0, 5, 0, 4])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -1947,6 +1954,7 @@ " plot_vector2d(vector, origin=origin)\n", "\n", "plt.axis([0, 5, 0, 4])\n", + "plt.gca().set_aspect(\"equal\")\n", "plt.grid()\n", "plt.show()" ] @@ -1988,9 +1996,12 @@ " plot_vector2d(vector_after, color=\"red\", linestyle=\"-\")\n", " plt.gca().add_artist(Polygon(P_before.T, alpha=0.2))\n", " plt.gca().add_artist(Polygon(P_after.T, alpha=0.3, color=\"r\"))\n", + " plt.plot(P_before[0], P_before[1], \"b--\", alpha=0.5)\n", + " plt.plot(P_after[0], P_after[1], \"r--\", alpha=0.5)\n", " plt.text(P_before[0].mean(), P_before[1].mean(), text_before, fontsize=18, color=\"blue\")\n", " plt.text(P_after[0].mean(), P_after[1].mean(), text_after, fontsize=18, color=\"red\")\n", " plt.axis(axis)\n", + " plt.gca().set_aspect(\"equal\")\n", " plt.grid()\n", "\n", "P_rescaled = 0.60 * P\n", @@ -2011,12 +2022,7 @@ { "cell_type": "code", "execution_count": 90, - "metadata": { - "collapsed": true, - "jupyter": { - "outputs_hidden": true - } - }, + "metadata": {}, "outputs": [], "source": [ "U = np.array([[1, 0]])" @@ -2035,7 +2041,7 @@ "metadata": {}, "outputs": [], "source": [ - "U.dot(P)" + "U @ P" ] }, { @@ -2052,7 +2058,7 @@ "outputs": [], "source": [ "def plot_projection(U, P):\n", - " U_P = U.dot(P)\n", + " U_P = U @ P\n", " \n", " axis_end = 100 * U\n", " plot_vector2d(axis_end[0], color=\"black\")\n", @@ -2060,10 +2066,12 @@ " plt.gca().add_artist(Polygon(P.T, alpha=0.2))\n", " for vector, proj_coordinate in zip(P.T, U_P.T):\n", " proj_point = proj_coordinate * U\n", - " plt.plot(proj_point[0][0], proj_point[0][1], \"ro\")\n", - " plt.plot([vector[0], proj_point[0][0]], [vector[1], proj_point[0][1]], \"r--\")\n", + " plt.plot(proj_point[0][0], proj_point[0][1], \"ro\", zorder=10)\n", + " plt.plot([vector[0], proj_point[0][0]], [vector[1], proj_point[0][1]],\n", + " \"r--\", zorder=10)\n", "\n", " plt.axis([0, 5, 0, 4])\n", + " plt.gca().set_aspect(\"equal\")\n", " plt.grid()\n", " plt.show()\n", "\n", @@ -2133,7 +2141,7 @@ "metadata": {}, "outputs": [], "source": [ - "V.dot(P)" + "V @ P" ] }, { @@ -2149,7 +2157,7 @@ "metadata": {}, "outputs": [], "source": [ - "P_rotated = V.dot(P)\n", + "P_rotated = V @ P\n", "plot_transformation(P, P_rotated, \"$P$\", \"$VP$\", [-2, 6, -2, 4], arrows=True)\n", "plt.show()" ] @@ -2207,7 +2215,7 @@ " [1, 1.5],\n", " [0, 1]\n", " ])\n", - "plot_transformation(P, F_shear.dot(P), \"$P$\", \"$F_{shear} P$\",\n", + "plot_transformation(P, F_shear @ P, \"$P$\", \"$F_{shear} P$\",\n", " axis=[0, 10, 0, 7])\n", "plt.show()" ] @@ -2229,7 +2237,7 @@ " [0, 0, 1, 1],\n", " [0, 1, 1, 0]\n", " ])\n", - "plot_transformation(Square, F_shear.dot(Square), \"$Square$\", \"$F_{shear} Square$\",\n", + "plot_transformation(Square, F_shear @ Square, \"$Square$\", \"$F_{shear} Square$\",\n", " axis=[0, 2.6, 0, 1.8])\n", "plt.show()" ] @@ -2251,7 +2259,7 @@ " [1.4, 0],\n", " [0, 1/1.4]\n", " ])\n", - "plot_transformation(P, F_squeeze.dot(P), \"$P$\", \"$F_{squeeze} P$\",\n", + "plot_transformation(P, F_squeeze @ P, \"$P$\", \"$F_{squeeze} P$\",\n", " axis=[0, 7, 0, 5])\n", "plt.show()" ] @@ -2269,7 +2277,7 @@ "metadata": {}, "outputs": [], "source": [ - "plot_transformation(Square, F_squeeze.dot(Square), \"$Square$\", \"$F_{squeeze} Square$\",\n", + "plot_transformation(Square, F_squeeze @ Square, \"$Square$\", \"$F_{squeeze} Square$\",\n", " axis=[0, 1.8, 0, 1.2])\n", "plt.show()" ] @@ -2291,7 +2299,7 @@ " [1, 0],\n", " [0, -1]\n", " ])\n", - "plot_transformation(P, F_reflect.dot(P), \"$P$\", \"$F_{reflect} P$\",\n", + "plot_transformation(P, F_reflect @ P, \"$P$\", \"$F_{reflect} P$\",\n", " axis=[-2, 9, -4.5, 4.5])\n", "plt.show()" ] @@ -2316,8 +2324,8 @@ " [1, -1.5],\n", " [0, 1]\n", "])\n", - "P_sheared = F_shear.dot(P)\n", - "P_unsheared = F_inv_shear.dot(P_sheared)\n", + "P_sheared = F_shear @ P\n", + "P_unsheared = F_inv_shear @ P_sheared\n", "plot_transformation(P_sheared, P_unsheared, \"$P_{sheared}$\", \"$P_{unsheared}$\",\n", " axis=[0, 10, 0, 7])\n", "plt.plot(P[0], P[1], \"b--\")\n", @@ -2328,7 +2336,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We applied a shear mapping on $P$, just like we did before, but then we applied a second transformation to the result, and *lo and behold* this had the effect of coming back to the original $P$ (we plotted the original $P$'s outline to double check). The second transformation is the inverse of the first one.\n", + "We applied a shear mapping on $P$, just like we did before, but then we applied a second transformation to the result, and *lo and behold* this had the effect of coming back to the original $P$ (I've plotted the original $P$'s outline to double check). The second transformation is the inverse of the first one.\n", "\n", "We defined the inverse matrix $F_{shear}^{-1}$ manually this time, but NumPy provides an `inv` function to compute a matrix's inverse, so we could have written instead:" ] @@ -2360,6 +2368,8 @@ " [0, 1, 1, 0, 0, 0.1, 1.1, 1.0, 1.1, 1.1, 1.0, 1.1, 0.1, 0, 0.1, 0.1],\n", " \"r-\")\n", "plt.axis([-0.5, 2.1, -0.5, 1.5])\n", + "plt.gca().set_aspect(\"equal\")\n", + "plt.grid()\n", "plt.show()" ] }, @@ -2382,7 +2392,7 @@ " [1, 0],\n", " [0, 0]\n", " ])\n", - "plot_transformation(P, F_project.dot(P), \"$P$\", \"$F_{project} \\cdot P$\",\n", + "plot_transformation(P, F_project @ P, \"$P$\", r\"$F_{project} \\cdot P$\",\n", " axis=[0, 6, -1, 4])\n", "plt.show()" ] @@ -2424,7 +2434,7 @@ " [np.cos(angle30)**2, np.sin(2*angle30)/2],\n", " [np.sin(2*angle30)/2, np.sin(angle30)**2]\n", " ])\n", - "plot_transformation(P, F_project_30.dot(P), \"$P$\", \"$F_{project\\_30} \\cdot P$\",\n", + "plot_transformation(P, F_project_30 @ P, \"$P$\", r\"$F_{project\\_30} \\cdot P$\",\n", " axis=[0, 6, -1, 4])\n", "plt.show()" ] @@ -2462,7 +2472,7 @@ "metadata": {}, "outputs": [], "source": [ - "F_shear.dot(LA.inv(F_shear))" + "F_shear @ LA.inv(F_shear)" ] }, { @@ -2506,7 +2516,7 @@ " [0, -2],\n", " [-1/2, 0]\n", " ])\n", - "plot_transformation(P, F_involution.dot(P), \"$P$\", \"$F_{involution} \\cdot P$\",\n", + "plot_transformation(P, F_involution @ P, \"$P$\", r\"$F_{involution} \\cdot P$\",\n", " axis=[-8, 5, -4, 4])\n", "plt.show()" ] @@ -2532,7 +2542,7 @@ "metadata": {}, "outputs": [], "source": [ - "F_reflect.dot(F_reflect.T)" + "F_reflect @ F_reflect.T" ] }, { @@ -2668,7 +2678,7 @@ " [0.5, 0],\n", " [0, 0.5]\n", " ])\n", - "plot_transformation(P, F_scale.dot(P), \"$P$\", \"$F_{scale} \\cdot P$\",\n", + "plot_transformation(P, F_scale @ P, \"$P$\", r\"$F_{scale} \\cdot P$\",\n", " axis=[0, 6, -1, 4])\n", "plt.show()" ] @@ -2721,7 +2731,7 @@ "metadata": {}, "outputs": [], "source": [ - "P_squeezed_then_sheared = F_shear.dot(F_squeeze.dot(P))" + "P_squeezed_then_sheared = F_shear @ (F_squeeze @ P)" ] }, { @@ -2737,7 +2747,7 @@ "metadata": {}, "outputs": [], "source": [ - "P_squeezed_then_sheared = (F_shear.dot(F_squeeze)).dot(P)" + "P_squeezed_then_sheared = F_shear @ F_squeeze @ P" ] }, { @@ -2752,16 +2762,11 @@ { "cell_type": "code", "execution_count": 122, - "metadata": { - "collapsed": true, - "jupyter": { - "outputs_hidden": true - } - }, + "metadata": {}, "outputs": [], "source": [ - "F_squeeze_then_shear = F_shear.dot(F_squeeze)\n", - "P_squeezed_then_sheared = F_squeeze_then_shear.dot(P)" + "F_squeeze_then_shear = F_shear @ F_squeeze\n", + "P_squeezed_then_sheared = F_squeeze_then_shear @ P" ] }, { @@ -2788,7 +2793,7 @@ "metadata": {}, "outputs": [], "source": [ - "LA.inv(F_shear.dot(F_squeeze)) == LA.inv(F_squeeze).dot(LA.inv(F_shear))" + "LA.inv(F_shear @ F_squeeze) == LA.inv(F_squeeze) @ LA.inv(F_shear)" ] }, { @@ -2855,7 +2860,7 @@ "metadata": {}, "outputs": [], "source": [ - "U.dot(np.diag(S_diag)).dot(V_T)" + "U @ np.diag(S_diag) @ V_T" ] }, { @@ -2880,7 +2885,7 @@ "metadata": {}, "outputs": [], "source": [ - "plot_transformation(Square, V_T.dot(Square), \"$Square$\", \"$V^T \\cdot Square$\",\n", + "plot_transformation(Square, V_T @ Square, \"$Square$\", r\"$V^T \\cdot Square$\",\n", " axis=[-0.5, 3.5 , -1.5, 1.5])\n", "plt.show()" ] @@ -2898,7 +2903,9 @@ "metadata": {}, "outputs": [], "source": [ - "plot_transformation(V_T.dot(Square), S.dot(V_T).dot(Square), \"$V^T \\cdot Square$\", \"$\\Sigma \\cdot V^T \\cdot Square$\",\n", + "plot_transformation(V_T @ Square, S @ V_T @ Square,\n", + " r\"$V^T \\cdot Square$\",\n", + " r\"$\\Sigma \\cdot V^T \\cdot Square$\",\n", " axis=[-0.5, 3.5 , -1.5, 1.5])\n", "plt.show()" ] @@ -2916,7 +2923,9 @@ "metadata": {}, "outputs": [], "source": [ - "plot_transformation(S.dot(V_T).dot(Square), U.dot(S).dot(V_T).dot(Square),\"$\\Sigma \\cdot V^T \\cdot Square$\", \"$U \\cdot \\Sigma \\cdot V^T \\cdot Square$\",\n", + "plot_transformation(S @ V_T @ Square, U @ S @ V_T @ Square,\n", + " r\"$\\Sigma \\cdot V^T \\cdot Square$\",\n", + " r\"$U \\cdot \\Sigma \\cdot V^T \\cdot Square$\",\n", " axis=[-0.5, 3.5 , -1.5, 1.5])\n", "plt.show()" ] @@ -3019,7 +3028,7 @@ " [ 10, 20, 30],\n", " [ 1, 2, 3],\n", " ])\n", - "np.trace(D)" + "D.trace()" ] }, { @@ -3042,7 +3051,7 @@ "metadata": {}, "outputs": [], "source": [ - "np.trace(F_project)" + "F_project.trace()" ] }, {