91影视

Consider three unit vectors ${\stackrel{鈯�}{\mathbf{v}}}_1,{\stackrel{鈯�}{\mathbf{v}}}_2$, and ${\stackrel{鈯�}{\mathbf{v}}}_3$in ${\mathit{R}}^{\mathbf{n}}$. We are told that . What are the possible values of role="math" localid="1660106727111" ${\stackrel{鈯�}{\mathbf{v}}}_1,{\stackrel{鈯�}{\mathbf{v}}}_2={\stackrel{鈯�}{\mathbf{v}}}_1,{\stackrel{鈯�}{\mathbf{v}}}_3=\mathbf{1}\mathbf{/}\mathbf{2}$? What could the angle between the vectors ${\stackrel{鈯�}{\mathit{v}}}_2$and ${\stackrel{鈯�}{\mathit{v}}}_3$ be? Give examples; draw sketches for the cases n = 2 and n = 3 .

Find the orthonormal basis ${\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{3}}$ of${\mathbf{鈻�}}^{\mathbf{3}}$ such that

role="math" localid="1660301787314" $\mathit{s}\mathit{p}\mathit{a}\mathit{n}\mathbf{\left(}{\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{p}\mathit{a}\mathit{n}\mathbf{\left(}\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\mathbf{\right)}$

and

$\mathit{s}\mathit{p}\mathit{a}\mathit{n}\mathbf{(}{\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{)}\mathbf{=}\mathit{s}\mathit{p}\mathit{a}\mathit{n}\mathbf{(}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\\ \mathbf{3}\end{array}\mathbf{\right]}\mathbf{,}\left[\begin{array}{c}1\\ 1\\ -1\end{array}\right]\mathbf{)}$

Consider a line Lin ${\mathit{R}}^{\mathbf{n}}$, spanned by a unit vector

$\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{=}\left[\begin{array}{c}{u}_1\\ u_2\\ .\\ .\\ .\\ u_n\end{array}\right]$

Consider the matrix Aof the orthogonal projection onto L. Describe the ijth entry of A, in terms of the components${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{i}}$of $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$.

TRUE OR FALSE

Determine whether the statement 鈥淭he matrix A is symmetric and matrix S is orthogonal, then matrix ${\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}$must be symmetric.

Can you find a line Lin ${\mathit{R}}^{\mathbf{n}}$and a vector$\stackrel{\mathbf{鈫�}}{\mathbf{x}}$ in ${\mathit{R}}^{\mathbf{n}}$such that$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{路}\mathbf{\text{pro}}{\mathbf{\text{j}}}_{\mathbf{L}}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$is negative? Explain, arguing algebraically.

TRUE OR FALSE? If matrix is A orthogonal, then the matrix${\mathit{A}}^{\mathbf{2}}$ must be orthogonal as well.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercise 1 through 14.

3.$\left[\left.\begin{array}{c}4\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}25\\ 0\\ -25\end{array}\right.\right]$

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? $\frac{\mathbf{1}}{\mathbf{3}}\left[\begin{array}{ccc}2& -2& 1\\ 1& 2& 2\\ 2& 1& -2\end{array}\right]$.

Consider a subspace V of${\mathbf{鈩�}}^{\mathbf{n}}{\mathbf{鈩�}}^{\mathbf{n}}$. Let${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{p}}$ be a basis of V and${\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{q}}$ a basis of. Is$\left\{{\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,....,{\stackrel{鈫�}{v}}_p,{\stackrel{鈫�}{w}}_1,{\stackrel{鈫�}{w}}_2,...,{\stackrel{鈫�}{w}}_q\right\}$ a basis of${\mathbf{鈩�}}^{\mathbf{n}}$? Explain.

Find the length of each of the vectors$\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$In exercises 1 through 3.

3.$\stackrel{\mathbf{鈬赌}}{\mathbf{v}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{2}\\ \mathbf{3}\\ \mathbf{4}\\ \mathbf{5}\end{array}\mathbf{\right]}$