README.md

Lab 12: Dot Product

In lecture, we've talked about what the dot product means to multiply a vector by a vector and a matrix by a vector. Here, we'll review that, and also learn what it means to multiply two matrices together.

1. Vector dot Vector

Complete the following function so that it compute the dot product of two vectors:

def v_dot_v(v1, v2):
    assert len(v1) == len(v2)
    total = 0
    for i in range(len(v1)):
        total += ????
    return total

a = np.array([100,10,1])
b = np.array([3,2,0])
v_dot_v(a, b) # should be 320

Although the vectors are one dimensional, we'll assume v1 is horizontal and v2 is vertical.

ANSWER

v1[i] * v2[i]

2. Matrix dot Vector

We can multiply a matrix by a vector by doing vector-by-vector multiplications, taking one row at a time from the matrix for the first vector. This will give us a vector containing output value per row in the matrix.

Complete the following function so that it computes the dot product of a matrix and a 1-dimensional vector (assumed to be vertical):

def m_dot_v(m, v):
    output = []
    for row in m:
        assert len(row) == len(v)
        output.append(????)
    return np.array(output)

A = np.array([
    [1,0,3],
    [0,2,3],
])
x = np.array([1,10,100])
m_dot_v(A, x) # should be [301, 320]

ANSWER

v_dot_v(row, v)

3. Matrix dot Matrix

We can multiply a matrix by a matrix by doing matrix-by-vector multiplications, taking one column at a time from the second matrix for the vector. Each of these multiplications gives an output vector -- arranging these output vectors as columns in an output matrix gives the result of the matrix multiplication.

Complete the following function so that it compute the dot product of a two matrices:

def m_dot_m(m1, m2):
    output_cols = []
    for col in m2.T:
        output_cols.append(????)
    return np.array(output_cols).T

A = np.array([
    [1,0],
    [1,2],
    [1,3],
    [0,5],
    [100,200],
])
B = np.array([
    [1,0,10],
    [0,1,1],
])
m_dot_m(A, B)

The result should be this:

array([[   1,    0,   10],
       [   1,    2,   12],
       [   1,    3,   13],
       [   0,    5,    5],
       [ 100,  200, 1200]])

ANSWER

m_dot_v(m1, col)