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Exercise

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.

Instructions

100 XP
  • 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.