Question

How does a linear transformation work?

Answer 1

Take any shape (such as the unit circle--a circle centered about the origin with radius 1). A linear transformation can stretch, translate (move around), rotate, flip, and do other movement.

y = G(x) where y is in \[\mathbb{R} ^{m}\] and and x is in \[\mathbb{R} ^{?}\], then the linear transformation G() maps x to y. This can be represented by a matrix A, such that y = Ax.

Matrix multiplication is another question. Does this answer your question? If not, could you give more detail into what your question is?