[선형대수학] 부분공간의 차원 (Dimension of Subspace)

$\underline{Thm}$

Let $W$ be a subspace of a finite-dimensional vector space $V$. Then $W$ is finite-dimensional, and $dim(W) \leq dim(V)$.

Furthermore, if $dim(W)=dim(V)$, then $W=V$.

 

$\underline{Proof}$

Write $dim\,V=n \in \mathbb{N}_{0}$.

If $W=\{ 0 \}$, then $W$ is finite-dimensional & $dim\,W=0 \leq$. Now, assume $W \neq \{0 \}$. Then $V \neq \{ 0 \}$ & $n \geq 1$.

We can choose non-zero $x_{1} \in W$ so that $\{ x_{1} \}$ is linear independent.

Choose a non-zero $x_{2} \in W \backslash \{ x_{1} \}$ so that $\{x_{1}, x_{2} \}$ is linear independent, if possible.

Continue this process until it's not possible. This process should stop after some finite step $k \in \mathbb{N}$, since otherwise we can choose $n+1$ linearly independent vectors $x_{1}, \cdots, x_{n+1} \in W \subset V$, so that $n+1 \leq n$ (by Coro (c)), a contradiction.

So, we can choose $x_{1}, \cdots, x_{k} \in W$ so that $\{x_{1}, \cdots, x_{k} \}$ is linear independent & $\{ x_{1}, \cdots, x_{k} \} \cup \{ v \}$ is linear dependent $\forall v \in W$.

By thm, $v \in span(\{x_{1}, \cdots, x_{k} \}) \forall v \in W$, i.e. $span(\{x_{1}, \cdots, x_{k} \}) =W$. By Coro (c), $k \leq n$.

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$\underline{Coro}$

Let $W,V$ be as above theorem. Then any basis for $W$ can be extended to a basis for $V$.

 

$\underline{Proof}$

Use Coro (c).