# Machine Learning Fundamentals - Linear Algebra - Exercise: Matrix and Vector Operations

## Introduction

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.

## Requirements

### Python Modules

# External Modules
import numpy as np

### Data

Given are the following matrices:

$$A = \begin{pmatrix} 4 & 4 & 5 \\ 2 & 1 & 7 \\ 4 & 8 & 3 \end{pmatrix} , B = \begin{pmatrix} 1 & 6 \\ 3 & 1 \\ 5 & 2 \end{pmatrix} , C = \begin{pmatrix} 1 & 4 & 4 \\ 3 & 1 & 2\\ 6 & 7 & 1 \end{pmatrix}$$

and following vectors:

$$\vec{x} = \begin{pmatrix} 9 \\ 5 \\ 7 \end{pmatrix} , \vec{y} = \begin{pmatrix} 3 \\ 1 \\ 5 \end{pmatrix}$$

# 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

Perform following calculations by hand or write some latex in that notebook.

1. $\vec{x} \cdot \vec{y}$ (dot or inner product)
2. $\vec{x} * \vec{y}^T$ (matrix product)
3. $A * B$
4. $B * A$
5. $A * C$
6. $C * A$
7. $(C^T * A^T)^T$
8. $A \circ C$ (Hadamard or Schur product)
9. $\left \langle A,C \right \rangle_F$ (Frobenius inner product)

## Implementation 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)

### Vector Subtraction

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)

### Scalar Multiplication

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)

### Dot Product

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)

### Matrix Multiplication

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)

### Transpose

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.

## Literature

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