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The singular value decomposition

Coordinates and transformations

The same arrow has different coordinates in different bases, and the same linear map has different matrices. This topic changes basis, writes the matrix of a map in any basis, and uses homogeneous coordinates to move and project pictures in computer graphics.

Lessons on this topic

Animation: the point (5, 1) and a grid sliding from the square grid to the grid of v1 = (1, −1) and v2 = (2, 1). Coordinates on the current grid (1, 2). A moves the point to (8.5, 3.5).Animation: the point (5, 1) and a grid sliding from the square grid to the grid of v1 = (1, −1) and v2 = (2, 1). Coordinates on the current grid (1, 2). A moves the point to (8.5, 3.5).

Intermediate · 25 minYou are hereRead

Change of basis

Keep a point still while the grid behind it changes, convert its coordinates between bases, and write a map in a basis where it is simple.

Animation: the coefficients (2, −3, 1, 4) of p(x) = 2 − 3x + x^2 + 4x^3 multiplied by the differentiation matrix D. The result (−3, 2, 12, 0) is p'(x) = −3 + 2x + 12x^2, drawn beside p; at x = −1, −0.5, 0, 0.5, 1 the slope of p equals the height of p'.Animation: the coefficients (2, −3, 1, 4) of p(x) = 2 − 3x + x^2 + 4x^3 multiplied by the differentiation matrix D. The result (−3, 2, 12, 0) is p'(x) = −3 + 2x + 12x^2, drawn beside p; at x = −1, −0.5, 0, 0.5, 1 the slope of p equals the height of p'.

Intermediate · 20 minYou are hereRead

The matrix of a linear map

Turn differentiation of polynomials into a matrix, build any linear map’s matrix column by column, and see the same map get different matrices in different bases.

Animation: a wireframe cube turned 53.13 degrees about the vertical axis, moved away from the eye, and projected onto a screen at distance 1. The move distance is 4. The cube is now a perspective drawing on the screen.Animation: a wireframe cube turned 53.13 degrees about the vertical axis, moved away from the eye, and projected onto a screen at distance 1. The move distance is 4. The cube is now a perspective drawing on the screen.

Advanced · 20 minYou are hereRead

Matrices in computer graphics

Add one coordinate so that a move becomes a matrix, chain moves, turns and scalings into one product, and project a 3-D object onto a screen.

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