Geometric Transformations with Matrices

Linear Transformations of the Plane

Every 2×2 matrix A defines a linear transformation of the plane: it maps each point (x, y) to a new point (x′, y′) via the rule [[x′],[y′]] = A[[x],[y]]. The transformation is linear because it preserves straight lines through the origin, and maps the origin to itself. This makes 2×2 matrices the perfect tool for describing rotations, reflections, dilations, and shears — the building blocks of geometric transformation.

The key insight is that a linear transformation is completely determined by where it sends the two standard basis vectors e₁ = [[1],[0]] and e₂ = [[0],[1]]. The first column of A is the image of e₁, and the second column is the image of e₂. This means that to find the matrix for any transformation, you just need to track what happens to (1,0) and (0,1).

Rotation Matrices

To rotate anticlockwise by angle θ about the origin, the matrix is R(θ) = [[cosθ, −sinθ], [sinθ, cosθ]]. You can derive this by tracking e₁ = (1,0) and e₂ = (0,1): rotating (1,0) by θ gives (cosθ, sinθ), which forms the first column; rotating (0,1) by θ gives (−sinθ, cosθ), which forms the second column.

Key special cases: R(90°) = [[0,−1],[1,0]]; R(180°) = [[−1,0],[0,−1]]; R(270°) = [[0,1],[−1,0]]; R(360°) = I. Notice that det(R(θ)) = cos²θ + sin²θ = 1, confirming that rotation preserves area and orientation.

Reflection Matrices

Reflections reverse orientation, so their determinant is always −1. The reflection in the x-axis sends (x,y) to (x,−y), giving matrix [[1,0],[0,−1]]. The reflection in the y-axis sends (x,y) to (−x,y): matrix [[−1,0],[0,1]]. The reflection in y = x swaps coordinates: (x,y) → (y,x), giving [[0,1],[1,0]].

Reflections are their own inverses: reflecting twice returns to the original. Algebraically, this means M² = I for any reflection matrix M, which you can verify directly. Geometrically, it means reflecting twice is the identity transformation.

Dilation Matrices

A dilation by factor k from the origin scales all distances by k. Its matrix is [[k,0],[0,k]] = kI. The determinant is k², confirming that areas are scaled by k² (a square of side 1 becomes a square of side k, with area k²). If k = −1, the dilation is the same as a 180° rotation.

Non-uniform scaling uses [[k,0],[0,m]] to scale by k in the x-direction and m in the y-direction. Such transformations are commonly used in computer graphics to stretch or compress images.

Composing Transformations

When we apply transformation T₁ followed by T₂, the overall effect is described by the matrix product T₂T₁. The right-to-left order of matrix multiplication mirrors the right-to-left reading of function composition: in f(g(x)), we apply g first then f. So to apply T₁ first, it goes on the right.

This non-commutativity of composition has real geometric consequences: rotating then reflecting generally gives a different result from reflecting then rotating. One order may give a reflection in a different line, or a rotation by a different angle. The matrix product captures this precisely.

The determinant of a composition equals the product of the individual determinants: det(T₂T₁) = det(T₂)det(T₁). So if you compose two rotations (det = 1 each), the result has det = 1 and is also a rotation. If you compose two reflections (det = −1 each), the result has det = 1 and is a rotation!