Linear Transformations
Linear Transformations
Definition of linear transformations, standard matrix representation, 2D transformations (rotation, reflection, scaling, shear), composition, invertible transformations.
Linear Transformations
Definition: A function T: ℝⁿ→ℝᵐ is a linear transformation if:
1. T(u+v) = T(u) + T(v) (additivity)
2. T(cu) = cT(u) (homogeneity)
Equivalently: T(cu+dv) = cT(u) + dT(v)
1. T(u+v) = T(u) + T(v) (additivity)
2. T(cu) = cT(u) (homogeneity)
Equivalently: T(cu+dv) = cT(u) + dT(v)
Matrix representation: Every linear transformation T: ℝⁿ→ℝᵐ can be written as T(x) = Ax for some m×n matrix A (the standard matrix of T).
Column j of A = T(eⱼ) where eⱼ is the j-th standard basis vector.
Column j of A = T(eⱼ) where eⱼ is the j-th standard basis vector.
Standard 2D Transformations
| Transformation | Matrix A | Effect |
|---|---|---|
| Rotation by θ (CCW) | [cos θ, −sin θ; sin θ, cos θ] | Rotates vectors anticlockwise by θ |
| Reflection about x-axis | [1,0; 0,−1] | Flips y-component sign |
| Reflection about y-axis | [−1,0; 0,1] | Flips x-component sign |
| Scaling | [s,0; 0,s] | Scales by factor s |
| Horizontal shear | [1,k; 0,1] | Shifts x by k×y |
| Projection onto x-axis | [1,0; 0,0] | Sets y=0 |
Rotation by 90° CCW: A = [[0,−1],[1,0]]
T([1,0]ᵀ) = [0,1]ᵀ ✓ (x-axis maps to y-axis)
T([0,1]ᵀ) = [−1,0]ᵀ ✓ (y-axis maps to negative x-axis)
T([1,0]ᵀ) = [0,1]ᵀ ✓ (x-axis maps to y-axis)
T([0,1]ᵀ) = [−1,0]ᵀ ✓ (y-axis maps to negative x-axis)
Composition of Transformations
T₂ ∘ T₁ corresponds to matrix product A₂A₁
(apply T₁ first, then T₂)
(apply T₁ first, then T₂)
Order matters: A₂A₁ ≠ A₁A₂ in general
Invertible Transformations
T is invertible iff det(A) ≠ 0. Then T⁻¹ corresponds to A⁻¹.
T⁻¹ undoes T: T⁻¹(T(x)) = x
T⁻¹ undoes T: T⁻¹(T(x)) = x
Kernel and Image
- Kernel (null space): ker(T) = {x | T(x) = 0} — inputs that map to zero
- Image (range): im(T) = {T(x) | x ∈ ℝⁿ} — all possible outputs
- Rank-nullity: dim(ker T) + dim(im T) = n