Linear Transformation - Yousef's Notes

A linear transformation is a function $T$ between two vector spaces $V$ and $W$ that satisfies: $$ T(a\mathbf{v} + b\mathbf{u}) = aT(\mathbf{v}) + bT(\mathbf{u}) $$

where $a$ and $b$ are scalars, and $\mathbf{v}$ and $\mathbf{u}$ are vectors in $V$. Defined by where it takes the Basis vectors.
In other words, a linear transformation preserves the operations of vector addition and scalar multiplication. The formula can be broken down into two parts:

Additivity: $T(\mathbf{v} + \mathbf{u}) = T(\mathbf{v}) + T(\mathbf{u})$
Homogeneity: $T(a\mathbf{v}) = aT(\mathbf{v})$ This formula ensures that the transformation is linear, meaning that it can be represented by a matrix. For example, a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ can be represented by a $2 \times 2$ matrix:

$$ T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} $$