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.
Cet exercice fait partie du cours
Linear Algebra for Data Science in R
Instructions
A,Bandbare 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.
Exercice interactif pratique
Essayez cet exercice en complétant cet exemple de 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