What matrices do to space — the geometry of linear maps.
The matrix doesn't just transform vectors — it transforms all of space.
Understanding transformations means understanding what gets stretched, compressed, rotated, reflected, or collapsed.
ARRANGEMENT: "Order matters."
A matrix encodes a transformation — a systematic rearrangement of space. Every point moves according to the same linear rule.
Definition: A function T: ℝⁿ → ℝᵐ is a linear transformation if:
Key insight: The columns of A are the images of the standard basis vectors.
Column 1 = where (1,0) goes. Column 2 = where (0,1) goes.
Matrix:
| Transformation | Matrix | det |
|---|---|---|
| Rotation by θ | [[cos θ, -sin θ], [sin θ, cos θ]] | 1 |
| Reflection across x | [[1, 0], [0, -1]] | -1 |
| Reflection across y | [[-1, 0], [0, 1]] | -1 |
| Scale by (sₓ, sᵧ) | [[sₓ, 0], [0, sᵧ]] | sₓsᵧ |
| Horizontal shear | [[1, k], [0, 1]] | 1 |
| Project onto x | [[1, 0], [0, 0]] | 0 |
Geometric meaning: The determinant measures how a transformation scales area (2D) or volume (3D).
det([[a,b],[c,d]]) = ad - bc = 1.65
2×2:
3×3 (cofactor expansion):
Kernel: ker(A) = {v : Av = 0} — vectors that get collapsed to zero
Image: im(A) = {Av : v ∈ ℝⁿ} — all possible outputs
Rank-Nullity Theorem: dim(ker) + dim(im) = n
BA =
Warning: The order is reversed!
Matrix:
det = ()
| Operation | det | Type |
|---|---|---|
| Identity E | +1 | Proper |
| Rotation Cₙ | +1 | Proper |
| Reflection σ | -1 | Improper |
| Inversion i | -1 | Improper |
| Improper rotation Sₙ | -1 | Improper |
| Feature | How to check |
|---|---|
| Preserves area? | |det| = 1 |
| Invertible? | det ≠ 0 |
| Preserves orientation? | det > 0 |
| Orthogonal? | AᵀA = I |
| Pure rotation? | Orthogonal and det = +1 |
| Has collapse? | det = 0 or nullity > 0 |
For each matrix, identify the geometric transformation:
(a) [[0, 1], [-1, 0]] (b) [[1, 0], [0, -1]] (c) [[3, 0], [0, 3]] (d) [[1, 2], [0, 1]]
(a) Rotation by -90° (or 270°). Check: (1,0) → (0,-1)
(b) Reflection across x-axis
(c) Uniform scaling by factor 3
(d) Horizontal shear with k = 2
Compute det and interpret geometrically:
(a) [[2, 0], [0, 3]] (b) [[1, 1], [2, 2]] (c) [[cos θ, -sin θ], [sin θ, cos θ]]
(a) det = 6. Area scales by 6. Stretches 2× horizontally, 3× vertically.
(b) det = 0. Singular — collapses space. Columns are parallel.
(c) det = cos²θ + sin²θ = 1. Area preserved — it's a rotation.
(a) Find the matrix for: reflect across x-axis, then rotate 90°.
(b) Find the matrix for: rotate 90°, then reflect across x-axis.
(c) Are they the same?
(a) R₉₀ × Reflₓ = [[0,-1],[1,0]][[1,0],[0,-1]] = [[0,1],[1,0]]
(b) Reflₓ × R₉₀ = [[1,0],[0,-1]][[0,-1],[1,0]] = [[0,-1],[-1,0]]
(c) Different! (a) is reflection across y=x; (b) is reflection across y=-x.
(a) Write the matrix for C₃ (120° rotation about z).
(b) Verify C₃³ = E.
(c) What is det(C₃)? Proper or improper?
(a) C₃ = [[cos120°, -sin120°], [sin120°, cos120°]] = [[-1/2, -√3/2], [√3/2, -1/2]]
(b) C₃³ = rotation by 360° = I ✓
(c) det(C₃) = 1. Proper rotation (preserves orientation).