11.1: Dot Product Review

We covered inner products a while ago. This assignment will extend the idea of inner products to matrix multiplication. As a reminder, Sections 1.4 of the Stephen Boyd and Lieven Vandenberghe Applied Linear algebra book covers the dot product. Here is a quick review:

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Given two vectors \(u\) and \(v\) in \(R^n\) (i.e. they have the same length), the “dot” product operation multiplies all of the corresponding elements and then adds them together. Ex:

\[u = [u_1, u_2, \dots, u_n] \nonumber \]

\[v = [v_1, v_2, \dots, v_n] \nonumber \]

\[u \cdot v = u_1 v_1 + u_2 v_2 + \dots + u_nv_n \nonumber \]

or:

\[u \cdot v = \sum^n_{i=1} u_i v_i \nonumber \]

This can easily be written as python code as follows:

u = [1,2,3]
v = [3,2,1]
solution = 0
for i in range(len(u)):
    solution += u[i]*v[i]
    
solution

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In numpy the dot product between two vectors can be calculated using the following built in function:

import numpy as np
np.dot([1,2,3], [3,2,1])

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Question

What is the dot product of any vector and the zero vector?

Question

What happens to the numpy.dot function if the two input vectors are not the same size?