Machine Learning Fundamentals - Linear Algebra - Exercise: Matrix and Vector Operations
Table of Contents
This exercise tests knowledge in basics in linear algebra. Knowledge about matrices, vectors, and their operations are essential in understanding more complex machine learning topics, like neural networks. Safe handling of domain-specific notation and concepts is therefore necessary.
- Chapter 2 of Deep Learning by Ian Goodfellow gives a brief introduction into the field
- Linear Algebra by Jim Hefferson is a open-source textbook with a lot of good exercises
- Introduction to Linear Algebra by Gilbert Strang ist a good domain specific textbook
- Coding the Matrix: Linear Algebra through Applications to Computer Science by Philip Klein is focused on a computer science viewpoint
# External Modules import numpy as np
Given are the following matrice:
and following vectors:
# Matrices A = ([4,4,5], [2,1,7], [4,8,3]) B = ([1,6], [3,1], [5,2]) C = ([1,4,4], [3,1,2], [6,7,1]) # Vectors x = (9,5,7) y = (3,1,5)
Pen and Paper Calculation
Solve following calculations by hand or write some latex in that notebook.
- (dot or inner product)
- (Hadamard or Schur product)
- (Frobenius inner product)
Implmentation of Basic Operations
Implement the following functions using the data structure
List only. The results should be the same as the corresponding Numpy implementation.
def vector_add(a, b): ''' Adds two given vectors a and b: https://en.wikipedia.org/wiki/Euclidean_vector params: a: A list representing vector a b: A list representing vector b returns: [x_1 + y_1, x_2 + y_2, ... , x_n + y_n] ''' raise NotImplementedError # Test np.testing.assert_array_almost_equal(vector_add(x,y), np.add(x,y), verbose=True)
def vector_sub(a, b): ''' Subtracts two given vectors a and b: https://en.wikipedia.org/wiki/Euclidean_vector params: a: A list representing vector a b: A list representing vector b returns: [x_1 - y_1, x_2 - y_2, ... , x_n - y_n] ''' raise NotImplementedError # Testing np.testing.assert_array_almost_equal(vector_sub(x,y), np.subtract(x,y), verbose=True)
def scalar_mul(r, A): ''' Multiply each element of a matrix or vector by a scalar 'r': https://en.wikipedia.org/wiki/Scalar_multiplication params: r: Scalar A: Vector or matrix returns: A vector or matrix with the same dimesion like 'A' but each element multiplied by r ''' raise NotImplementedError # Testing sca = 3 np.testing.assert_array_almost_equal(scalar_mul(sca,A), np.multiply(sca,A), verbose=True)
def vec_dot(a, b): ''' Sum of the product of corresponding elements: https://en.wikipedia.org/wiki/Dot_product params: a: Vector b: Vector returns: x_1 * y_1 + x_2 * y_2 + ... + x_n * y_n ''' raise NotImplementedError # Testing np.testing.assert_array_almost_equal(vec_dot(x,y), np.dot(x,y), verbose=True)
def matrix_mult(A, B): ''' Computes the product of two matrices: https://en.wikipedia.org/wiki/Matrix_multiplication params: A: Matrix with dimensions NxP B: Matrix with dimensions PxM returns: NxM matrix with each element c_i_j = a_i_1 * b_1_j + ... + a_i_p * b_p_j ''' raise NotImplementedError # Testing np.testing.assert_array_almost_equal(matrix_mult(A,B), np.matmul(A,B), verbose=True)
def matrix_transpose(A): ''' Flips the matrix over its diagonal, switches the row and column indices of the matrix: https://en.wikipedia.org/wiki/Transpose params: A: Matrix returns: Transpose A^T of given matrix A ''' raise NotImplementedError # Testing np.testing.assert_array_almost_equal(matrix_transpose(A), np.transpose(A), verbose=True)
Summary and Outlook
This exercise covered basic operations on vectors and matrices. If the exercise was too complicated, consider the sources mentioned above for a recap.
Notebook License (CC-BY-SA 4.0)
The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).
Exercise: Matrix and Vector Operations
by Benjamin Voigt
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.
Code License (MIT)
The following license only applies to code cells of the notebook.
Copyright 2018 Benjamin Voigt
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
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