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A matrix moves the plane
Watch a matrix move the whole grid, read where the two unit arrows land, and compute a matrix times a vector two ways.
Vectors and matrices
Matrices as maps
A matrix moves every point of the plane at once, and its columns say where the two unit arrows land. This topic reads rotations, shears, reflections and projections from their matrices, then multiplies matrices by doing one map after another.
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![Animation: two copies of a picture; the left one is turned a quarter turn then sheared, the right one sheared then turned. Now the left one has e1 at (1, 1) and e2 at (−1, 0), the right one e1 at (0, 1) and e2 at (−1, 1). The single matrices are [1, −1; 1, 0] on the left and [0, −1; 1, 1] on the right.](/thumbs/matrix-multiplication-dark.webp)
![Animation: two copies of a picture; the left one is turned a quarter turn then sheared, the right one sheared then turned. Now the left one has e1 at (1, 1) and e2 at (−1, 0), the right one e1 at (0, 1) and e2 at (−1, 1). The single matrices are [1, −1; 1, 0] on the left and [0, −1; 1, 1] on the right.](/thumbs/matrix-multiplication-light.webp)
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