Linear Combinations and Span

Scale some vectors and add them up. Everything you can reach that way is their span — the whole plane, a flat sheet in 3D, or just a line when the vectors line up.

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Two vectors, a = (2, 1) in amber and b = (−1, 1) in purple. A linear combination scales each by a number and adds: c₁a + c₂b. Which points can we reach?

a =

b =

(c₁, c₂) =

The combination appears once the tour reaches it.

det [a b] ≠ 0 means a and b point in different directions, so their span is the whole plane; det = 0 means they share a line.

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Linear Combinations and Span

See also

Linear Combinations and Span

See also