Subspace

Definition

Let $V$ be a vector space over the field $F$. A subspace of $V$ is a subset $W\subset V$ which itself is a vector space over $F$ with the same operations defined on $V$.

Observations

Observation 1

In any vector space $V$, $V$ and $\{0\}$ are subspaces of $V$. The latter is called the zero subspace of $V$.

Theorems

Theorem 1 *¹

A non-empty subset $W$ of $V$ is a subspace of $V$ if and only if $\forall \mathbf{x},\mathbf{y}\in W,\forall c\in F:$ $c\mathbf{x}+\mathbf{y}\in W$.

Proof

$(\Rightarrow )$ Trivial.

$(\Leftarrow )$ Since $W\ne \varnothing$, $\forall \mathbf{v}\in W:$ $(-1)\mathbf{v}+\mathbf{v}=0\in W$.

$\forall \mathbf{w}\in W,\forall c\in F:$ $c\mathbf{w}=c\mathbf{w}+0\in W$. In particular, $(-1)\mathbf{w}=-\mathbf{w}\in W$.

Finally, $\forall \mathbf{v},\mathbf{w}\in W:$ $\mathbf{v}+\mathbf{w}=1\mathbf{v}+\mathbf{w}\in W$.

Therefore, $W$ is a subspace of $V$. \tag*{$||$}

Theorem 2 *¹

The intersection of any collection of subspaces of a vector space $V$ is a subspace of $V$.

Proof

Let $\{W\_a\}$ be the collection of subspaces of $V$, and let $W=\cap \_aW\_a$. Since $0\in W\_a$ for all $W\_a$, $0\in W$ and thus $W$ is nonempty.

Let $\mathbf{x},\mathbf{y}\in W$ and $c$ be an arbitrary scalar. By definition of $W$, $\mathbf{x},\mathbf{y}\in W\_a$ for all $W\_a$ and since they are all subspaces of $V$, $c\mathbf{x}+\mathbf{y}\in W\_a$ for all $W\_a$. Thus $c\mathbf{x}+\mathbf{y}\in W$. \tag*{$||$}

Footnotes

1. Linear Algebra, Second Edition, by Kenneth Hoffman and Ray Kunze ↩ ↩²