[DEFINITIONS] Subspaces

Subspace

Let $V$ be a vector space over $F$ . A subset $S\subseteq V$ is a subspace of $V$ when it is a vector space over the same field $F$ and uses the addition and scalar multiplication inherited from $V$ . A proper subspace is a subspace that is not equal to $V$ .

Two elements of $S$ are added as they are in $V$ and scalar multiplication uses the same scalars. The sets $\{0\}$ and $V$ are always subspaces of $V$ . When $m\leq n$ , the standard embedding

(x_1,\ldots,x_m)\longmapsto(x_1,\ldots,x_m,\underbrace{0,\ldots,0}_{n-m})

identifies $\mathbb R^m$ with the coordinate subspace $\{(x_1,\ldots,x_m,0,\ldots,0)\}\subseteq\mathbb R^n$ . For $n\ge2$ , a plane through the origin in $\mathbb R^n$ is another example.

[1]:Vector space
[2]:Kernel, image and rank
[3]:A projection onto a subspace
[4]:Seen Questions
[5]:The span is a subspace
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Definitions from Subspaces

Subspace

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