Exercise

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.

Instructions

100 XP
  • 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.