91影视

Independent vectors and spanning vectors in a subspace of${\mathbf{鈩�}}^{\mathbf{n}}$

Consider a subspace $\mathit{V}$of ${\mathbf{鈩�}}^{\mathbf{n}}$with$\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{=}\mathit{m}$.

1. We can find at most $\mathit{m}$linearly independent vectors in role="math" localid="1664290582958" $\mathit{V}$.
2. We need at least $\mathit{m}$vectors to span $\mathit{V}$.
3. If m vectors in $\mathit{V}$are linearly independent, then they form a basis of $\mathit{V}$.
4. If m vectors in $\mathit{V}$span $\mathit{V}$, then they form a basis of $\mathit{V}$.

Given that,$V$and$W$are the subspaces of ${\mathrm{鈩�}}^{10}$i.e.,$W,V鈭�{\mathrm{鈩�}}^{10}$.

$$\begin{gathered} \dim V=6 \\ \dim W=7 \end{gathered}$$

The possible dimensions of$V鈭�W$ is to be found.

Since,$W,V鈭�{\mathrm{鈩�}}^{10}$ therefore, there exist following results:

$\dim {\mathrm{鈩�}}^{10}=10$

role="math" localid="1664291194065" $$\begin{gathered} \begin{array}{rcl}& & \left.\max \left\{\dim \left(W+V\right)\right\}=10 \\ \min \left\{\dim \left(W+V\right)\right\}=7\right\}\dim \left(W+V\right)=\left\{10,9,8,7\right\}\end{array} \end{gathered}$$

From exercise we have:

$$\begin{gathered} \dim \left(V鈭�W\right)=\dim V+\dim W-\dim \left(V+W\right) \\ =6+7-\left\{10,9,8,7\right\} \\ =13-\left\{10,9,8,7\right\} \\ =\left\{3,4,5,6\right\} \end{gathered}$$

Therefore, the possible dimension of $\dim \left(V鈭�W\right)=\left\{3,4,5,6\right\}$.