08-linear-algebra.md

1.8 Linear Algebra Refresher

Slides

Notes

Linear Algebra Refresher

• Vector operations
• Multiplication
  • Vector-vector multiplication
  • Matrix-vector multiplication
  • Matrix-matrix multiplication
• Identity matrix
• Inverse

Vector operations

u = np.array([2, 7, 5, 6])
v = np.array([3, 4, 8, 6])

# addition 
u + v

# subtraction 
u - v

# scalar multiplication 
2 * v

Multiplication

Vector-vector multiplication

def vector_vector_multiplication(u, v):
    assert u.shape[0] == v.shape[0]
    
    n = u.shape[0]
    
    result = 0.0

    for i in range(n):
        result = result + u[i] * v[i]
    
    return result

Matrix-vector multiplication

def matrix_vector_multiplication(U, v):
    assert U.shape[1] == v.shape[0]
    
    num_rows = U.shape[0]
    
    result = np.zeros(num_rows)
    
    for i in range(num_rows):
        result[i] = vector_vector_multiplication(U[i], v)
    
    return result

Matrix-matrix multiplication

def matrix_matrix_multiplication(U, V):
    assert U.shape[1] == V.shape[0]
    
    num_rows = U.shape[0]
    num_cols = V.shape[1]
    
    result = np.zeros((num_rows, num_cols))
    
    for i in range(num_cols):
        vi = V[:, i]
        Uvi = matrix_vector_multiplication(U, vi)
        result[:, i] = Uvi
    
    return result

Identity matrix

I = np.eye(3)

Inverse

V = np.array([
    [1, 1, 2],
    [0, 0.5, 1], 
    [0, 2, 1],
])
inv = np.linalg.inv(V)

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• Notes from Peter Ernicke - Part 1/3
• Notes from Peter Ernicke - Part 2/3
• Notes from Peter Ernicke - Part 3/3

Links

• Notebook from the video
• Get a visual understanding of matrix multiplication
• Overview of matrix multiplication functions in python/numpy

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