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[core/

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Core Concept Map Language

Theorems from Subspaces

All theorems from Subspaces.

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[THEOREM]

Subspace test

Let $S$ be a subset of a vector space $V$ over $F$ , with the operations inherited from $V$ . Then $S$ is a subspace of $V$ if and only if:

$S$ is non-empty;
$S$ is closed under vector addition: $u,v\in S$ implies $u+v\in S$ ;
$S$ is closed under scalar multiplication: $u\in S$ and $\lambda\in F$ imply $\lambda u\in S$ .

Non-emptiness and scalar closure supply the zero vector and additive inverses, so these three conditions are enough.

[1]:Subspace
[2]:Vector space
[3]:The kernel is a subspace
[4]:The image is a subspace
[5]:a set of polynomial divisors
[6]:subspaces of $M_{2\times2}(\mathbb R)$
8 more

This page builds on Vector spaces.

It leads to Seen Questions, Kernel, image and rank (Linear transformations), Spans and Seen Questions (Linear transformations).

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