Linear and multilinear algebra; matrix theory Theorem

Every spanning set contains a basis

Statement

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $S \subseteq V$ be a spanning set of $V$ . Then there is a subset $B \subseteq S$ that is a basis of $V$ .

Remarks

For finite $S$ , remove from $S$ , one at a time, a vector that is a linear combination of the other vectors. When no such vector remains, the set is linearly independent and still spans $V$ . In general, take a maximal linearly independent subset of $S$ , by Zorn's lemma. For example, $\{(1, 0), (0, 1), (1, 1)\}$ spans $\R^2$ and contains the basis $\{(1, 0), (1, 1)\}$ .

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Vector space definition
Linear combination definition
Linear subspace definition
Linear independence definition
Linear span definition
Spanning set definition
Basis definition
Every spanning set contains a basis statement

Tag	`06MRNP4BNT`
URI	`m2://06MRNP4BNT`
Kind	statement, theorem
Writing	W0 stub
MSC	15A03
Status	This copy of the site has no build data. Run `m2 status 06MRNP4BNT`.
Uses	Basis, Spanning set

Every spanning set contains a basis

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here