Linear Transformations

A linear transformation is a function $T: V \to W$ preserving vector addition and scalar multiplication:

$$T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v}) \quad \text{and} \quad T(\alpha\vec{u}) = \alpha\,T(\vec{u})$$

Combined, this requires $T(\alpha\vec{u} + \beta\vec{v}) = \alpha\,T(\vec{u}) + \beta\,T(\vec{v})$.

Every linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ is represented by an $m\times n$ matrix $A$ via $T(\vec{x}) = A\vec{x}$. Crucial rule: The columns of $A$ are the exact images of the standard basis vectors.

Pitfall: Guessing entries without testing basis vectors is how rotations get mistaken for shears.

Kernel, Image, and Matrices

The kernel $\ker(T) = \{\vec{v} : T(\vec{v})=\vec{0}\}$ is the null space. The image $\text{Im}(T) = \{T(\vec{v})\}$ is the column space.

$T$ is injective when $\ker(T) = \{\vec{0}\}$, and surjective when $\text{Im}(T) = W$.

Determinant: $|\det A|$ scales areas. $\det A < 0$ reverses orientation.