Matrix Multiplication - Order Matters

In the last lesson, we studied how matrices act on vectors (stretches, shrinkages, reflections, rotations, etc.) and transform vectors into new vectors.

The successive application of these matrices can act as complex transformations, but because matrix multiplication is not commutative, the order of these transformations matter.

The matrix with R output

> A
          [,1]       [,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068  0.7071068

represents rotation of a 2-dimensional vector by 45 degrees counterclockwise.

The matrix

> B
     [,1] [,2]
[1,]    1    0
[2,]    0   -1

represents a reflection about the x (first) axis.

This exercise is part of the course

Linear Algebra for Data Science in R

View Course

Exercise instructions

A, B and b are loaded for you. Compute the products \(AB\) and \(BA\) and show that these two actions are not commutative.
Apply both of these products by the vector b <- c(1,1) to further confirm.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Multiply A by B
A%*%___

# Multiply A on the right of B
___%*%A

# Multiply the product of A and B by the vector b
A%*%B%*%___

# Multiply A on the right of B, and then by the vector b
B%*%___%*%b

Edit and Run Code