elenasm7 / dsc-2-13-10-linalg-dot-product-properties-lab-nyc-ds-career-012819

Properties of Dot Product - Lab

Introduction

In this lab we shall look at some interesting properties of a Dot product type matrix multiplication. Understanding these properties will become useful as we move forward with machine learning advanced linear algebra. The lab will require you to calculate results to provide a proof for these properties.

Objectives

You will be able to:

• Understand and analytically explain Distributive, Commutative and Associative properties of dot product

Instructions

• For each property, create suitably sized matrices with random data and 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 - Matrices multiplication is distributive

Prove that A.(B+C)=A.B+A.C

# Your code here

Associative Property - Matrices multiplication is associative

Prove that A.(B.C)=(A.B).C

# Your code here 

Commutative Property - Matrix multiplication is NOT commutative

Prove that for matrices, A.B ≠ B.A

# Your code here 

Commutative Property - vector multiplication IS commutative

Prove that for vectors, x^T . y = y^T . x

Note: superscipt^T denotes the transpose we saw earlier

# Your code here 

and finally

Simplification of the matrix product

Prove that (A.B)^T = B^T . A^T

# Your code here 

Summary

So now we have seen enough matrix algebra to help us solve a problem of linear equations as we saw earlier in this section. We shall see how to do this next.