{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "### Author: Fatemeh yaghoobi\n",
    "#### September 2022\n",
    "\n",
    "#### The aim of this notebook is to provide the basic and fundemental linear algebra operations in numPy which will be useful to know for \"Basics of sensor fusion\" course.\n",
    "\n",
    "#### REFERECES:\n",
    "\n",
    "NumPy for MATLAB users: the NumPy documentation which help MATLAB users get started with NumPy [here](https://numpy.org/devdocs/user/numpy-for-matlab-users.html).\n",
    "\n",
    "Python basics [notebook](https://colab.research.google.com/github/data-psl/lectures2020/blob/master/notebooks/01_python_basics.ipynb) prepared by [Mathieu Blonde](https://mblondel.org/).\n",
    "\n",
    "Matlab vs. Python -- A rebuttal [here](https://johnfoster.pge.utexas.edu/blog/posts/matlab-vs-python/#disqus_thread).\n",
    "\n",
    "Matrix Operations in NumPy vs. Matlab [here](https://mccormickml.com/2019/10/28/matrix-operations-in-numpy-vs-matlab/)."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### To use NumPy in your program, you need to import it as follows (remember to pip install numpy first)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "import numpy as np"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# Vectors and vector operations\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# A vector that has 4 elements\n",
    "v1 = np.array([3, 4, 5])\n",
    "v1"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# multipying vector with a scalar\n",
    "# When a vector is multiplied by a scalar, \n",
    "#each element of the vector is multiplied with the scalar\n",
    "\n",
    "v1*2\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# add (subtract) 2 vectors: Element-wise summation\n",
    "#and subtraction on vectors are done with standard math operations.\n",
    "v1 = np.array([3, 4, 5])\n",
    "v2 = np.array([2, 0, -2])\n",
    "print(\"v1+v2 = \", v1 + v2)\n",
    "print(\"v1-v2 = \", v1 - v2)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# dot product: The dot product of two vectors \n",
    "#is the sum of the products of elements with regards to position\n",
    "v1 = np.array([3, 4, 5])\n",
    "v2 = np.array([2, 0, -2])\n",
    "\n",
    "np.dot(v1, v2)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "# Matrix and matrix operations"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# A matrix is a 2-dimensional array.\n",
    "my_matrix = np.array([[1, 2, 3], [4, 5, 6]])\n",
    "\n",
    "print(my_matrix)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Arrays of all zeros"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "#zero matrix\n",
    "zero_matrix = np.zeros((2, 3))\n",
    "zero_matrix"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Arrays of ones"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# ones matrix\n",
    "ones_matrix = np.ones((3,4))\n",
    "ones_matrix"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Shape of matices:\n",
    "We define the shape of a matrix in terms of the number of rows and columns"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "print(\"shape of my_matrix:\", my_matrix.shape)\n",
    "print(\"shape of ones_matrix:\", ones_matrix.shape)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Accessing specific elements, rows, columns"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "a = np.array([[1,2,3],[4,5,6], [7,8,9]])\n",
    "a"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "a.shape"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# We can access individual elements of a 2d-array using two indices\n",
    "a[3,1]"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Note: NumPy numbers indices from 0. When we have 3 rows, the indices are 0, 1 and 2"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# We can also access rows\n",
    "a_row = a[0]  # or a[0, :] # First row\n",
    "a_row\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# and columns\n",
    "a_column = a[:, 2] # Last column\n",
    "a_column\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## matrix-matrix multiplication"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### `numpy.dot`"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "M1 = np.array([[1, 2], [3, 4]])\n",
    "M2 = np.array([[5, 7], [6, 8]])\n",
    "print(M1)\n",
    "print()\n",
    "print(M2)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "np.dot(M1, M2)\n",
    "\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "Each element of the product matrix is a dot product of a row in first matrix and a column in the second matrix. "
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## `@`: The built-in operator to do matrix-vector or matrix-matrix multiplication"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "M1 @ M2"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "# Transpose"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "print(M1)\n",
    "print(M1.T)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    " ## Let's once again look at vectors in numpy"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# ndarray treats vectors as 1-dimensional\n",
    "v1\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "v1.shape"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### In Matlab, a vector is a 2-dimensional object – it is either a column vector (e.g., [3 x 1]) or a row vector (e.g., [1 x 3])."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# transpose\n",
    "print(v1.T)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "#### Numpy doesn’t naturally distinguish between a row vector and a column vector.\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "M3 = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])\n",
    "M3"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# Multiply v1 with M3."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "np.dot(M3, v1)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "In above operation, numPy assumes that you know what you're doing and that \n",
    "`v1` is a column vector with shape [3 x 1]\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "np.dot(v1, M3)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "In above operation, numPy assumes that you know what you're doing and that \n",
    "`v1` is a row vector with shape [1 x 3]"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Safe method: force vectors to have two dimensions \n",
    "    \n",
    "    \n",
    "    "
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# call the reshape function on the 1d vectors in NumPy \n",
    "print(v1.shape)\n",
    "print(np.reshape(v1, (3,1)).shape)\n",
    "print(np.reshape(v1, (1,3)).shape)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# put the vector’s index inside square brackets\n",
    "print(v1.shape)\n",
    "print(v1[:, None].shape)\n",
    "print(v1[None, :].shape)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "#adding a second set of brackets to explicitly add the second dimension to the row vector. \n",
    "v1_new = np.array([[1, 2, 3]])\n",
    "print(v1_new.shape)\n",
    "print(v1_new.T.shape)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Other useful functions for creating arrays:\n",
    "\n",
    "We saw `np.zeros`, `np.ones`"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# create a range of values using\n",
    "np.arange(5)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# or specifying the starting point\n",
    "np.arange(3, 5)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# creating linearly spaced values in an interval. \n",
    "np.linspace(0, 1, 10)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# creating random integres\n",
    "np.random.randint(low = 1, high = 6, size = (2,3))"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# creating random samples from a uniform distribution over [0,1)\n",
    "np.random.rand(2,4)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "np.dot(np.array([2, 2, -1]), np.array([-1,2,2]))"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "np.linalg.norm(np.array([3, 4]))"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# create a range of values using\n",
    "np.arange(5)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# or specifying the starting point\n",
    "np.arange(3, 5)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# creating linearly spaced values in an interval. \n",
    "np.linspace(0, 1, 10)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# creating random integres\n",
    "np.random.randint(low = 1, high = 6, size = (2,3))"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# creating random samples from a uniform distribution over [0,1)\n",
    "np.random.rand(2,4)\n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  }
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