Linear Transformations

A function $T: V \to W$ is a linear transformation if for all $\vec{u},\vec{v}\in V$ and $\alpha\in\mathbb{R}$:

$$T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v}) \qquad T(\alpha\vec{u}) = \alpha\,T(\vec{u})$$

Every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be represented by an $m\times n$ matrix.

Common transformations in $\mathbb{R}^{2}$

Kernel and image

• $\ker(T) = \{\vec{v} \in V : T(\vec{v})=\vec{0}\}$ — the null space.
• $\text{Im}(T) = \{T(\vec{v}) : \vec{v}\in V\}$ — the column space.
• $T$ is injective $\iff$ $\ker(T) = \{\vec{0}\}$.
• $T$ is surjective $\iff$ $\text{Im}(T) = W$.

Physics link: Rotations, reflections, and Lorentz boosts are all linear transformations. The matrix representation lets us compose them by simple multiplication.

Common pitfall: A matrix’s columns are the images of the basis vectors — that is the decoding key. Guessing a transformation from its entries without this reading is how rotations get mistaken for shears.

A linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ satisfies:

$$T(\alpha\vec{u} + \beta\vec{v}) = \alpha\,T(\vec{u}) + \beta\,T(\vec{v})$$

Every linear map is represented by a matrix $A$ such that $T(\vec{x}) = A\vec{x}$. The columns of $A$ are the images of the standard basis vectors.

Common transformations: rotations, reflections, scaling, shearing.

$|\det A|$ measures how the map scales areas. If $\det A < 0$, it reverses orientation.