Linear Transformations

Preserving Structure

A linear transformation is a function $T$ that preserves vector operations: $T(u + v) = T(u) + T(v)$ and $T(cv) = c T(v)$. Transforming a sum equals summing the transforms.

Every linear transformation between finite spaces is a matrix: $T(v) = Av$. To find $A$, evaluate $T$ on the basis vectors; their outputs form the columns.

Common pitfall: Assuming every function is linear. A translation $T(v) = v + b$ ($b \neq 0$) is not linear because it moves the origin. Linearity strictly demands $T(\mathbf{0}) = \mathbf{0}$.

Composition and Anatomy

Because transformations are matrices, composing them means multiplying their matrices. Order matters: rotate-then-reflect $\neq$ reflect-then-rotate.

Two structural quantities define a transformation:

• Kernel: The set of vectors sent to the zero vector.
• Image: The set of all reachable output vectors.

A transformation is invertible if and only if its matrix has a nonzero determinant.

Linear transformations drive computing: 3D graphics pipelines chain matrix transformations, and neural network layers apply linear transformations followed by nonlinear activations.