Matrix Multiplication as a Transformation
Matrices can be viewed as a way to transform collections of vectors into other vectors.
These transformations can take many forms, but the simplest ones in two dimensions are stretches or shrinkages (in either coordinate), reflections (e.g. about the x-axis, y-axis, origin, the line y = x), and rotations (clockwise, counter-clockwise).
Multiplication of a vector by a matrix is accomplished using the %*% command.
This exercise is part of the course
Linear Algebra for Data Science in R
Exercise instructions
- Use matrix multiplication in R to show that multiplication by the matrix \(A\) with R output:
> A
[,1] [,2]
[1,] 4 0
[2,] 0 1
stretches the x (first) component of the vector b <- c(1,1) by a factor of four.
- Show that multiplication by the matrix \(B\) with R output:
> B
[,1] [,2]
[1,] 1 0
[2,] 0 2/3
shrinks the y (second) component of the vector b <- c(1,1) by 33 percent.
Hands-on interactive exercise
Have a go at this exercise by completing this sample code.
# Multiply A by b
A ___ b
# Multiply B by b
___ ___ b