How does a matrix transform vectors?

This post gives an introduction to geometric spatial transformations through an example.

The figure above shows a vector V1 expressed as (x1, y1) , and another vector V2 which is obtained by rotating V1 anticlockwise.

Assume the vector length is r, we can express x1 and y1 as


Then for V2, we have


and


If we expand the vectors to a 3-D space, we will have


Based on results above, the relationship between V1 and V2 can be expressed as