Linear Transformations

Linear transformations in ℝ²

A map \(T:\mathbb{R}^2\!\to\mathbb{R}^2\) is linear when it satisfies additivity and homogeneity:

• \(T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})\)
• \(T(c\mathbf{v})=cT(\mathbf{v})\)

Every such map is captured by a 2×2 matrix \(A\) via \(T(\mathbf{v})=A\mathbf{v}\).

Common forms

Scaling \(\begin{bmatrix}s_x&0\\0&s_y\end{bmatrix}\) — stretches axes independently.

Rotation \(\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\) — preserves distances.

Shear \(\begin{bmatrix}1&k\\0&1\end{bmatrix}\) — slants along one axis.

Reflection \(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\) — flips across y-axis.

Determinant

\(\det(A)=ad-bc\) measures signed area scaling. \(\det=0\) collapses the plane; \(\det<0\) reverses orientation.