6. Linear transformations

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

– Arthur Cayley (1821 - 1895)

In this section, we introduce the basic objects of linear algebra, namely linear transformations. The main idea of this section is that every linear transformation can be represented by a matrix, and every matrix represents a linear transformation. One of the purposes of this section is to understand this fundamental correspondence. Under this correspondence, multiplying matrices is the same as composing linear transformations.

We will mostly focus on the 2-dimensional (${ℝ}^2$) case, and occasionally mention the $3$-dimensional case (${ℝ}^3$). The concepts in this section can be generalised to higher dimensional spaces; this will be done in the module MATH220 (linear algebra).