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[core/

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Core Concept Map Language

Methods from Spans

All methods from Spans.

[METHOD]

Test membership in a span

To decide whether $u\in\mathbb R^m$ lies in $\operatorname{span}\{v_1,\ldots,v_n\}$ , form $A=[\,v_1\ \cdots\ v_n\,]$ and solve

Ac=u.

A solution for the scalar coefficients exists if and only if $u$ belongs to the span, equivalently to $\operatorname{col}(A)$ .

Use row reduction on the augmented matrix $[A\mid u]$ . A contradictory row means that $u$ is not in the span.

[1]:Span
[2]:Column space
[3]:A vector outside a span
[4]:Three vectors spanning $\mathbb R^3$
[5]:Membership in a polynomial span
[6]:A set that does not span $P_2$

This page builds on Vector spaces and Subspaces.

It leads to Linear independence, Basis and dimension, Seen Questions, Data fitting and polynomial interpolation and Coordinates and bases.

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Test membership in a span