When Students have taken their first Linear Algebra course, they should have been taught, that a column vector can be multiplied by a matrix, to result in a column vector. They should also have been taught, that when matrices are used to multiply a column vector more than once, to result in a final column vector, the operation proceeds from right to left, and that the matrices which do so can themselves be multiplied, as the operation is associative. This multiplication can result in one matrix, as long as the number of rows of the first (right-hand) matrix is always equal to the number of columns in the second (left-hand) matrix.

One subject which does not usually get taught in beginning Linear Algebra courses, is that when the vectors are part of the same coordinate system, the matrix is equally capable of defining a rotation. What tends to get taught first, is transformations that appear linear and parallel.

The worksheet below is intended to show, that the correct choice of elements in a matrix, can also define a rotation:

(Edit 12/30/2018, 10h30 : )

The work-sheet has been updated, also to give a hint as to how 3D, Euler Angles may be translated into a 3×3 matrix.

(Edit 1/4/2019, 8h05 : )

I have now created a version of the same work-sheet, which can be viewed on a smart-phone. Please excuse the formatting errors which result:

Dirk