91影视

Consider a subspace V of ${\mathrm{鈩�}}^n$such that $\stackrel{鈫�}{x}鈭�V$and $\stackrel{鈫�}{y}鈭�V{}^{鈯�}$.

If the vector$\stackrel{\mathbf{鈫�}}{\mathbf{x}}$is perpendicular $\stackrel{\mathbf{鈫�}}{\mathbf{y}}$then $\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{-}\stackrel{\mathbf{鈫�}}{\mathbf{y}}\mathbf{=}\mathbf{0}$.

By the definition, the dot product of the vectors$\stackrel{鈫�}{x}$ and$\stackrel{鈫�}{y}$ is defined as follows.

$\stackrel{鈫�}{x}-\stackrel{鈫�}{y}=0$

Similarly, forrole="math" localid="1659438267836" $\stackrel{鈫�}{x}鈭�{\left(V^{鈯�}\right)}^{鈯�}$ and $\stackrel{鈫�}{y}鈭�V^{鈯�}$, the dot product of the vectors$\stackrel{鈫�}{x}$ and$\stackrel{鈫�}{y}$ is defined as follows.

$\stackrel{鈫�}{x}-\stackrel{鈫�}{y}=0$

Therefore, the setV is subset of ${\left(V^{鈯�}\right)}^{鈯�}$.

Theorem: Property of the orthogonal compliment.

Consider a subspace Vof ${\mathbf{R}}^{\mathbf{n}}$.

a. The orthogonal complement role="math" localid="1659438558378" ${\mathit{V}}^{\mathbf{鈯�}}$of is a subspace of ${\mathbf{R}}^{\mathbf{n}}$.

b. The intersection of Vand ${\mathbf{V}}^{\mathbf{鈯�}}$consist of the zero vector: $\mathit{V}\mathbf{鈭�}{\mathit{V}}^{\mathbf{鈯�}}\mathbf{=}\left\{\stackrel{鈫�}{0}\right\}$.

c. $\mathbf{dim}\mathbf{\left(}\mathbf{V}\mathbf{\right)}\mathbf{+}\mathbf{dim}\mathbf{\left(}{\mathbf{V}}^{\mathbf{鈯�}}\mathbf{\right)}\mathbf{=}\mathbf{n}$

d.$\mathbf{(}{\mathbf{V}}^{\mathbf{鈯�}}{\mathbf{)}}^{\mathbf{鈯�}}\mathbf{=}\mathbf{V}$

By the theorem, the equation for the set V is defined as follows.

$\dim \mathrm{V}+\dim {\mathrm{V}}^{鈯�}=\mathrm{n}$

By the theorem, the equation for the set $V^{鈯�}$is defined as follows.

$\dim {\mathrm{V}}^{鈯�}+\dim ({\mathrm{V}}^{鈯�}{)}^{鈯�}=\mathrm{n}$

Compare the equations ${\mathrm{dimV}}^1+\dim \left({\mathrm{V}}^1\right)=\mathrm{n}$and $\dim \mathrm{V}+\dim {\mathrm{V}}^{鈯�}=\mathrm{n}$as follows.

$$\begin{gathered} \dim \mathrm{V}+{\mathrm{dimV}}^{鈯�}={\mathrm{dimV}}^{鈯�}+\dim {\left({\mathrm{V}}^{鈯�}\right)}^{鈯�} \\ \dim \mathrm{V}=\dim {\left({\mathrm{V}}^{鈯�}\right)}^{鈯�} \end{gathered}$$

As $\dim \mathrm{V}=\dim ({\mathrm{V}}^{鈯�}{)}^{鈯�}$and$V鈯�(V^{鈯�}{)}^{鈯�}$ implies $V=(V^{鈯�}{)}^{鈯�}$.

Hence, the values $V=(V^{鈯�}{)}^{鈯�}$for subsetV of ${\mathrm{鈩�}}^{\mathrm{n}}$.