added in linear algebra contents

akhilvasvani · Mar 19, 2019 · 3c9ef3b · 3c9ef3b
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diff --git a/...d Tensors/.ipynb_checkpoints/2.1 Scalars, Vectors, Matrices, and Tensors-checkpoint.ipynb b/...d Tensors/.ipynb_checkpoints/2.1 Scalars, Vectors, Matrices, and Tensors-checkpoint.ipynb
diff --git a/... Scalars, Vectors, Matrices and Tensors/2.1 Scalars, Vectors, Matrices, and Tensors.ipynb b/... Scalars, Vectors, Matrices and Tensors/2.1 Scalars, Vectors, Matrices, and Tensors.ipynb
diff --git a/...alars, Vectors, Matrices and Tensors/images/dimensions-transposition-matrix.png b/...alars, Vectors, Matrices and Tensors/images/dimensions-transposition-matrix.png
diff --git a/Chapters/2.01 Scalars, Vectors, Matrices and Tensors/images/matrix-addition.png b/Chapters/2.01 Scalars, Vectors, Matrices and Tensors/images/matrix-addition.png
diff --git a/...lars, Vectors, Matrices and Tensors/images/non-squared-matrix-transposition.png b/...lars, Vectors, Matrices and Tensors/images/non-squared-matrix-transposition.png
diff --git a/...1 Scalars, Vectors, Matrices and Tensors/images/scalar-vector-matrix-tensor.png b/...1 Scalars, Vectors, Matrices and Tensors/images/scalar-vector-matrix-tensor.png
diff --git a/...1 Scalars, Vectors, Matrices and Tensors/images/square-matrix-transposition.png b/...1 Scalars, Vectors, Matrices and Tensors/images/square-matrix-transposition.png
diff --git a/...ers/2.01 Scalars, Vectors, Matrices and Tensors/images/vector-transposition.png b/...ers/2.01 Scalars, Vectors, Matrices and Tensors/images/vector-transposition.png
diff --git a/...ices and Vectors/.ipynb_checkpoints/2.2 Multiplying Matrices and Vectors-checkpoint.ipynb b/...ices and Vectors/.ipynb_checkpoints/2.2 Multiplying Matrices and Vectors-checkpoint.ipynb
diff --git a/Chapters/2.02 Multiplying Matrices and Vectors/2.2 Multiplying Matrices and Vectors.ipynb b/Chapters/2.02 Multiplying Matrices and Vectors/2.2 Multiplying Matrices and Vectors.ipynb
diff --git a/Chapters/2.02 Multiplying Matrices and Vectors/images/dot-product.png b/Chapters/2.02 Multiplying Matrices and Vectors/images/dot-product.png
diff --git a/Chapters/2.02 Multiplying Matrices and Vectors/images/plot-linear-equation.png b/Chapters/2.02 Multiplying Matrices and Vectors/images/plot-linear-equation.png
diff --git a/...Multiplying Matrices and Vectors/images/system-linear-equations-matrix-form.png b/...Multiplying Matrices and Vectors/images/system-linear-equations-matrix-form.png
diff --git a/...nd Inverse Matrices/.ipynb_checkpoints/2.3 Identity and Inverse Matrices-checkpoint.ipynb b/...nd Inverse Matrices/.ipynb_checkpoints/2.3 Identity and Inverse Matrices-checkpoint.ipynb
diff --git a/Chapters/2.03 Identity and Inverse Matrices/2.3 Identity and Inverse Matrices.ipynb b/Chapters/2.03 Identity and Inverse Matrices/2.3 Identity and Inverse Matrices.ipynb
diff --git a/Chapters/2.03 Identity and Inverse Matrices/images/identity-matrix.png b/Chapters/2.03 Identity and Inverse Matrices/images/identity-matrix.png
diff --git a/...ar Dependence and Span/.ipynb_checkpoints/2.4 Linear Dependence and Span-checkpoint.ipynb b/...ar Dependence and Span/.ipynb_checkpoints/2.4 Linear Dependence and Span-checkpoint.ipynb
diff --git a/Chapters/2.04 Linear Dependence and Span/2.4 Linear Dependence and Span.ipynb b/Chapters/2.04 Linear Dependence and Span/2.4 Linear Dependence and Span.ipynb
diff --git a/Chapters/2.04 Linear Dependence and Span/images/intersection-2-planes-line.png b/Chapters/2.04 Linear Dependence and Span/images/intersection-2-planes-line.png
diff --git a/...rs/2.04 Linear Dependence and Span/images/number-solutions-system-equations.png b/...rs/2.04 Linear Dependence and Span/images/number-solutions-system-equations.png
diff --git a/...04 Linear Dependence and Span/images/overdetermined-system-linear-equations.png b/...04 Linear Dependence and Span/images/overdetermined-system-linear-equations.png
diff --git a/.../2.04 Linear Dependence and Span/images/python-adding-vectors-combination-1.png b/.../2.04 Linear Dependence and Span/images/python-adding-vectors-combination-1.png
diff --git a/...rs/2.04 Linear Dependence and Span/images/python-adding-vectors-combination.png b/...rs/2.04 Linear Dependence and Span/images/python-adding-vectors-combination.png
diff --git a/Chapters/2.04 Linear Dependence and Span/images/python-one-equation.png b/Chapters/2.04 Linear Dependence and Span/images/python-one-equation.png
diff --git a/...r Dependence and Span/images/python-system-equations-with-linear-dependence.png b/...r Dependence and Span/images/python-system-equations-with-linear-dependence.png
diff --git a/Chapters/2.04 Linear Dependence and Span/images/python-three-equations.png b/Chapters/2.04 Linear Dependence and Span/images/python-three-equations.png
diff --git a/Chapters/2.04 Linear Dependence and Span/images/python-two-equations-1.png b/Chapters/2.04 Linear Dependence and Span/images/python-two-equations-1.png
diff --git a/...ers/2.04 Linear Dependence and Span/images/python-two-equations-no-solution.png b/...ers/2.04 Linear Dependence and Span/images/python-two-equations-no-solution.png
diff --git a/Chapters/2.04 Linear Dependence and Span/images/python-two-equations.png b/Chapters/2.04 Linear Dependence and Span/images/python-two-equations.png
diff --git a/Chapters/2.04 Linear Dependence and Span/images/python-two-vectors.png b/Chapters/2.04 Linear Dependence and Span/images/python-two-vectors.png
diff --git a/Chapters/2.04 Linear Dependence and Span/images/representing-features.png b/Chapters/2.04 Linear Dependence and Span/images/representing-features.png
diff --git a/...4 Linear Dependence and Span/images/underdetermined-system-linear-equations.png b/...4 Linear Dependence and Span/images/underdetermined-system-linear-equations.png
diff --git a/Chapters/2.05 Norms/.ipynb_checkpoints/2.5 Norms-checkpoint.ipynb b/Chapters/2.05 Norms/.ipynb_checkpoints/2.5 Norms-checkpoint.ipynb
diff --git a/Chapters/2.05 Norms/2.5 Norms.ipynb b/Chapters/2.05 Norms/2.5 Norms.ipynb
diff --git a/Chapters/2.05 Norms/images/l1-norm.png b/Chapters/2.05 Norms/images/l1-norm.png
diff --git a/Chapters/2.05 Norms/images/l2-norm.png b/Chapters/2.05 Norms/images/l2-norm.png
diff --git a/Chapters/2.05 Norms/images/squared-L2-Norm.png b/Chapters/2.05 Norms/images/squared-L2-Norm.png
diff --git a/...and Vectors/.ipynb_checkpoints/2.6 Special Kinds of Matrices and Vectors-checkpoint.ipynb b/...and Vectors/.ipynb_checkpoints/2.6 Special Kinds of Matrices and Vectors-checkpoint.ipynb
diff --git a/....06 Special Kinds of Matrices and Vectors/2.6 Special Kinds of Matrices and Vectors.ipynb b/....06 Special Kinds of Matrices and Vectors/2.6 Special Kinds of Matrices and Vectors.ipynb
diff --git a/...pecial Kinds of Matrices and Vectors/images/diagonal-and-symmetric-matrices.png b/...pecial Kinds of Matrices and Vectors/images/diagonal-and-symmetric-matrices.png
diff --git a/Chapters/2.06 Special Kinds of Matrices and Vectors/images/diagonal-matrix.png b/Chapters/2.06 Special Kinds of Matrices and Vectors/images/diagonal-matrix.png
diff --git a/Chapters/2.06 Special Kinds of Matrices and Vectors/images/orthogonal-matrix.png b/Chapters/2.06 Special Kinds of Matrices and Vectors/images/orthogonal-matrix.png
diff --git a/Chapters/2.06 Special Kinds of Matrices and Vectors/images/symmetric-matrix.png b/Chapters/2.06 Special Kinds of Matrices and Vectors/images/symmetric-matrix.png
diff --git a/Chapters/2.07 Eigendecomposition/.ipynb_checkpoints/2.7 Eigendecomposition-checkpoint.ipynb b/Chapters/2.07 Eigendecomposition/.ipynb_checkpoints/2.7 Eigendecomposition-checkpoint.ipynb
diff --git a/Chapters/2.07 Eigendecomposition/2.7 Eigendecomposition.ipynb b/Chapters/2.07 Eigendecomposition/2.7 Eigendecomposition.ipynb
diff --git a/Chapters/2.07 Eigendecomposition/images/output_59_0.png b/Chapters/2.07 Eigendecomposition/images/output_59_0.png
diff --git a/Chapters/2.07 Eigendecomposition/images/quadratic-functions-indefinite-form.png b/Chapters/2.07 Eigendecomposition/images/quadratic-functions-indefinite-form.png
diff --git a/...s/2.07 Eigendecomposition/images/quadratic-functions-negative-definite-form.png b/...s/2.07 Eigendecomposition/images/quadratic-functions-negative-definite-form.png
diff --git a/...s/2.07 Eigendecomposition/images/quadratic-functions-positive-definite-form.png b/...s/2.07 Eigendecomposition/images/quadratic-functions-positive-definite-form.png
diff --git a/... Value Decomposition/.ipynb_checkpoints/2.8 Singular Value Decomposition-checkpoint.ipynb b/... Value Decomposition/.ipynb_checkpoints/2.8 Singular Value Decomposition-checkpoint.ipynb
diff --git a/Chapters/2.08 Singular Value Decomposition/2.8 Singular Value Decomposition.ipynb b/Chapters/2.08 Singular Value Decomposition/2.8 Singular Value Decomposition.ipynb
diff --git a/Chapters/2.08 Singular Value Decomposition/images/SVD_image_dim.png b/Chapters/2.08 Singular Value Decomposition/images/SVD_image_dim.png
diff --git a/...osition/images/dimensions-reconstruction-image-singular-value-decomposition.png b/...osition/images/dimensions-reconstruction-image-singular-value-decomposition.png
diff --git a/....08 Singular Value Decomposition/images/non-square-matrix-change-dimensions.png b/....08 Singular Value Decomposition/images/non-square-matrix-change-dimensions.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/output_35_7.png b/Chapters/2.08 Singular Value Decomposition/images/output_35_7.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/rescaled-circle-rotated.png b/Chapters/2.08 Singular Value Decomposition/images/rescaled-circle-rotated.png
diff --git a/... Decomposition/images/singular-value-decomposition-understanding-dimensions.png b/... Decomposition/images/singular-value-decomposition-understanding-dimensions.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/singular-value-decomposition.png b/Chapters/2.08 Singular Value Decomposition/images/singular-value-decomposition.png
diff --git a/...rs/2.08 Singular Value Decomposition/images/transformation-vector-by-matrix.png b/...rs/2.08 Singular Value Decomposition/images/transformation-vector-by-matrix.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/unit-circle-transformation.png b/Chapters/2.08 Singular Value Decomposition/images/unit-circle-transformation.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/unit-circle-transformation1.png b/Chapters/2.08 Singular Value Decomposition/images/unit-circle-transformation1.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/unit-circle.png b/Chapters/2.08 Singular Value Decomposition/images/unit-circle.png
diff --git a/Chapters/2.08 Singular Value Decomposition/images/unit-vectors-rotation.png b/Chapters/2.08 Singular Value Decomposition/images/unit-vectors-rotation.png
diff --git a/Chapters/2.08 Singular Value Decomposition/test_svd.jpg b/Chapters/2.08 Singular Value Decomposition/test_svd.jpg
diff --git a/...ose Pseudoinverse/.ipynb_checkpoints/2.9 The Moore-Penrose Pseudoinverse-checkpoint.ipynb b/...ose Pseudoinverse/.ipynb_checkpoints/2.9 The Moore-Penrose Pseudoinverse-checkpoint.ipynb
diff --git a/Chapters/2.09 The Moore-Penrose Pseudoinverse/2.9 The Moore-Penrose Pseudoinverse.ipynb b/Chapters/2.09 The Moore-Penrose Pseudoinverse/2.9 The Moore-Penrose Pseudoinverse.ipynb
diff --git a/Chapters/2.09 The Moore-Penrose Pseudoinverse/images/dataset-representation.png b/Chapters/2.09 The Moore-Penrose Pseudoinverse/images/dataset-representation.png
diff --git a/Chapters/2.09 The Moore-Penrose Pseudoinverse/images/linear-regression-r.png b/Chapters/2.09 The Moore-Penrose Pseudoinverse/images/linear-regression-r.png
diff --git a/...e Moore-Penrose Pseudoinverse/images/overdetermined-system-equations-python.png b/...e Moore-Penrose Pseudoinverse/images/overdetermined-system-equations-python.png
diff --git a/Chapters/2.10 The Trace Operator/.ipynb_checkpoints/2.10 The Trace Operator-checkpoint.ipynb b/Chapters/2.10 The Trace Operator/.ipynb_checkpoints/2.10 The Trace Operator-checkpoint.ipynb
@@ -0,0 +1,383 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": true,
+    "scrolled": false
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "import seaborn as sns"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Populating the interactive namespace from numpy and matplotlib\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Plot style\n",
+    "sns.set()\n",
+    "%pylab inline\n",
+    "pylab.rcParams['figure.figsize'] = (4, 4)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\newcommand\\norm[1]{\\left\\lVert#1\\right\\rVert} \n",
+    "\\DeclareMathOperator{\\Tr}{Tr}\n",
+    "\\newcommand\\bs[1]{\\boldsymbol{#1}}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Introduction\n",
+    "\n",
+    "This chapter is very light! I can assure you that you will read it in 1 minute! It is nice after the last two chapters that were quite big! We will see what is the Trace of a matrix. It will be needed for the last chapter on the Principal Component Analysis (PCA)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 2.10 The Trace Operator"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<img src=\"images/trace-matrix.png\" width=\"200\" alt=\"Calculating the trace of a matrix\" title=\"Calculating the trace of a matrix\">\n",
+    "<em>The trace of matrix</em>\n",
+    "\n",
+    "\n",
+    "The trace is the sum of all values in the diagonal of a square matrix.\n",
+    "\n",
+    "$$\n",
+    "\\bs{A}=\n",
+    "\\begin{bmatrix}\n",
+    "    2 & 9 & 8 \\\\\\\\\n",
+    "    4 & 7 & 1 \\\\\\\\\n",
+    "    8 & 2 & 5\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\mathrm{Tr}(\\bs{A}) = 2 + 7 + 5 = 14\n",
+    "$$\n",
+    "\n",
+    "Numpy provides the function `trace()` to calculate it:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[2, 9, 8],\n",
+       "       [4, 7, 1],\n",
+       "       [8, 2, 5]])"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A = np.array([[2, 9, 8], [4, 7, 1], [8, 2, 5]])\n",
+    "A"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "14"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A_tr = np.trace(A)\n",
+    "A_tr"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "GoodFellow et al. explain that the trace can be used to specify the Frobenius norm of a matrix (see [2.5](https://hadrienj.github.io/posts/Deep-Learning-Book-Series-2.5-Norms/)). The Frobenius norm is the equivalent of the $L^2$ norm for matrices. It is defined by:\n",
+    "\n",
+    "$$\n",
+    "\\norm{\\bs{A}}_F=\\sqrt{\\sum_{i,j}A^2_{i,j}}\n",
+    "$$\n",
+    "\n",
+    "Take the square of all elements and sum them. Take the square root of the result. This norm can also be calculated with:\n",
+    "\n",
+    "$$\n",
+    "\\norm{\\bs{A}}_F=\\sqrt{\\Tr({\\bs{AA}^T})}\n",
+    "$$\n",
+    "\n",
+    "We can check this. The first way to compute the norm can be done with the simple command `np.linalg.norm()`:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "17.549928774784245"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "np.linalg.norm(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The Frobenius norm of $\\bs{A}$ is 17.549928774784245.\n",
+    "\n",
+    "With the trace the result is identical:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "17.549928774784245"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "np.sqrt(np.trace(A.dot(A.T)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Since the transposition of a matrix doesn't change the diagonal, the trace of the matrix is equal to the trace of its transpose:\n",
+    "\n",
+    "$$\n",
+    "\\Tr(\\bs{A})=\\Tr(\\bs{A}^T)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Trace of a product\n",
+    "\n",
+    "$$\n",
+    "\\Tr(\\bs{ABC}) = \\Tr(\\bs{CAB}) = \\Tr(\\bs{BCA})\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "### Example 1.\n",
+    "\n",
+    "Let's see an example of this property.\n",
+    "\n",
+    "$$\n",
+    "\\bs{A}=\n",
+    "\\begin{bmatrix}\n",
+    "    4 & 12 \\\\\\\\\n",
+    "    7 & 6\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\bs{B}=\n",
+    "\\begin{bmatrix}\n",
+    "    1 & -3 \\\\\\\\\n",
+    "    4 & 3\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\bs{C}=\n",
+    "\\begin{bmatrix}\n",
+    "    6 & 6 \\\\\\\\\n",
+    "    2 & 5\n",
+    "\\end{bmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "531"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A = np.array([[4, 12], [7, 6]])\n",
+    "B = np.array([[1, -3], [4, 3]])\n",
+    "C = np.array([[6, 6], [2, 5]])\n",
+    "\n",
+    "np.trace(A.dot(B).dot(C))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "531"
+      ]
+     },
+     "execution_count": 15,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "np.trace(C.dot(A).dot(B))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "531"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "np.trace(B.dot(C).dot(A))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\bs{ABC}=\n",
+    "\\begin{bmatrix}\n",
+    "    360 & 432 \\\\\\\\\n",
+    "    180 & 171\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\bs{CAB}=\n",
+    "\\begin{bmatrix}\n",
+    "    498 & 126 \\\\\\\\\n",
+    "    259 & 33\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\bs{BCA}=\n",
+    "\\begin{bmatrix}\n",
+    "    -63 & -54 \\\\\\\\\n",
+    "    393 & 594\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\Tr(\\bs{ABC}) = \\Tr(\\bs{CAB}) = \\Tr(\\bs{BCA}) =  531\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# References\n",
+    "\n",
+    "[Trace (linear algebra) - Wikipedia](https://en.wikipedia.org/wiki/Trace_(linear_algebra))\n",
+    "\n",
+    "[Numpy Trace operator](https://docs.scipy.org/doc/numpy/reference/generated/numpy.trace.html)"
+   ]
+  }
+ ],
+ "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": "3.6.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}