dot-product

Dot Product

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

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)