Matrix Transformations

Also called

matrix transformation · linear transformation · 2 × 2 matrix · scaling matrix · rotation matrix · shear matrix · columns of a matrix · unit vectors i and j · determinant as area · area scale factor · inverse of a matrix · singular matrix · identity matrix · आव्यूह · सारणिक · व्युत्क्रम आव्यूह

Topics

• A 2 × 2 matrix [a b; c d] moves every point of the plane: (x, y) goes to (ax + by, cx + dy).
• Its columns show where the unit arrows land: î goes to (a, c) and ĵ goes to (b, d).
• [2 0; 0 1.5] stretches 2 times across and 1.5 times up, so every area becomes 3 times as large.
• The rotation [cos θ −sin θ; sin θ cos θ] turns the plane by θ; the shear [1 1; 0 1] slides it sideways.
• The determinant ad − bc is the area scale factor: the unit square becomes a parallelogram of area ad − bc.
• Rotations and shears have determinant 1, so the L keeps its area of 3 squares.
• A matrix with determinant 0 squashes the plane onto a line; nothing can undo that, so it has no inverse.
• The inverse undoes the matrix: A⁻¹A = AA⁻¹ = I, and det(A⁻¹) = 1 / det(A).

About this model

A GLB file of about 138 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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