Linear Algebra

Span

The span of a set of vectors is everywhere you can reach by scaling and adding them. Two vectors usually fill the whole plane, but line them up and their span collapses to a single line. Drag the vectors and the target to feel independence, dependence, and coordinates.

Combination v = a·u₁ + b·u₂

a 1.00

b 1.00

Show combination lattice Solve a, b to hit the target

Presets

Spanthe whole plane

Determinant [u₁ u₂]—

Combination v—

Target reachable?—

Drag the tips of u₁ (indigo) and u₂ (amber), or the whitetarget. The teal dot is the current combination a·u₁ + b·u₂.

What to observe

Slide a and b: the teal dot sweeps out everything the two vectors can build. With independent u₁, u₂ the reachable set is thewhole plane (shaded), so every target has exactly one pair (a, b).
Now drag u₂ until it lies along u₁ (or use the Dependent preset). The shaded region collapses to a line: the vectors are linearly dependent, the determinant hits 0, and targets off that line becomeunreachable.
Turn on Solve and drag the target around. The a, b that hit it are its coordinates in the u₁, u₂ basis, read straight off the lattice. Watch them explode as the basis gets close to degenerate.

Shortcuts: space run/pause · s step · r reset · f fullscreen