Linear Algebra

Linear Transformations

A linear map is fully described by where it sends the basis vectors. Drag the tips of î and ĵ and the whole grid follows.

îĵv and AvF and its imageunit square, area = det A

Drag the tips of î and ĵ, or the ring marked v

det of the frame shown—

—

Transformation matrix

1.400.700.201.10

a 1.40

b 0.70

c 0.20

d 1.10

Presets

Path from I to A

View

Sweep I → A, t 1.00

Show

The figure, an FThe original grid and FA vector to drag

Values

î →—

ĵ →—

v—

Av—

turned so far—

What to observe

Pick Shear and drag the ring v. Its image is x steps along Aî plus y steps along Aĵ (the dashed legs): where the basis lands decides everything.
Animate Rotate 90° on the Turn + stretch path. The frame at time t is R(tθ) blended with A's stretch, so the grid and the F turn rigidly and det stays 1 the whole way.
Same matrix on the Straight blend (1−t)I + tA. It cuts the corner instead of turning: halfway the grid has shrunk to det 0.5, then grows back. Both paths end on the same A.
The shaded square's area and the directions that only stretch have their own pages: Determinantand Eigenvectors.

Shortcuts:space animate · f fullscreen