learn-co-students / dsc-linalg-dot-product-properties-lab-onl01-dtsc-pt-012120

Properties of Dot Product - Lab

Introduction

In this lab, you'll be practicing some interesting properties of a dot product-type matrix multiplication. Understanding these properties will become useful as you study machine learning. The lab will require you to calculate results to provide a proof for these properties.

Objectives

In this lab you will:

• Demonstrate the distributive, commutative, and associative property of dot products
• Use the transpose method to transpose Numpy matrices
• Compute the dot product for matrices and vectors

Instructions

• For each property, create suitably sized matrices with random data to prove the equations
• Ensure that size/dimension assumptions are met while performing calculations (you'll see errors otherwise)
• Calculate the LHS and RHS for all equations and show if they are equal or not

Distributive Property - matrix multiplication IS distributive

Prove that $A \cdot (B+C) = (A \cdot B + A \cdot C) $

# Your code here

Associative Property - matrix multiplication IS associative

Prove that $A \cdot (B \cdot C) = (A \cdot B) \cdot C $

# Your code here 

Commutative Property - matrix multiplication is NOT commutative

Prove that for matrices, $A \cdot B \neq B \cdot A $

# Your code here 

Commutative Property - vector multiplication IS commutative

Prove that for vectors, $x^T \cdot y = y^T \cdot x$

Note: superscipt^T denotes the transpose we saw earlier

# Your code here 

Simplification of the matrix product

Prove that $ (A \cdot B)^T = B^T \cdot A^T $

# Your code here 

Summary

You've seen enough matrix algebra by now to solve a problem of linear equations as you saw earlier. You'll now see how to do this next.