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Consider an inner product $<v,w>$ in a space V. Let wbe a fixed element of V. Is the transformation $\mathit{T}\left(v\right)\mathbf{=}<v,w>$from Vto R linear? What is its image? Give ageometric interpretation of its kernel.

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

9.$\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ 9\\ -5\\ 3\end{array}\right]$

For each pair of vectors $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$and $\stackrel{\mathit{鈫�}}{v}$listed in Exercises 7 through 9, determine whether the angle $\mathit{胃}$between $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$and$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$ is acute, obtuse, or right.

9.$\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{=}\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right]\mathbf{,}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\left[\begin{array}{c}3\\ 4\\ 5\\ 3\end{array}\right]$.

If the$\mathbf{n}\mathbf{脳}\mathbf{n}$ matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?${\mathbf{B}}^{\mathbf{-}\mathbf{1}}$.

If $\stackrel{\mathbf{鈬赌}}{\mathbf{u}}$is a unit vector in ${\mathbf{R}}^{\mathbf{n}}$and $\mathbf{L}\mathbf{=}\mathbf{span}\left(\stackrel{鈬赌}{u}\right)$, then ${\mathbf{proj}}_{\mathbf{L}}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\left(\stackrel{鈬赌}{x}.\stackrel{鈬赌}{u}\right)\stackrel{\mathbf{鈬赌}}{\mathbf{u}}\mathbf{.}$for all the vectors$\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{R}}^{\mathbf{n}}$.

Consider the linear system$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ , where

$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{3}\\ \mathbf{2}& \mathbf{6}\end{array}\mathbf{\right]}$and $\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{10}\\ \mathbf{20}\end{array}\mathbf{\right]}$.

a. Draw a sketch showing the following subsets of ${\mathit{鈩�}}^{\mathbf{2}}$:

• The kernel of$A$ , and${\mathbf{\left(}\mathbf{k}\mathbf{e}\mathbf{r}\mathbf{A}\mathbf{\right)}}^{\mathbf{鈯�}}$
• The image of ${\mathit{A}}^{\mathbf{T}}$
• The solution set$\mathit{S}$of the system $\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$

b.What relationship do you observe between$\mathbf{\left(}\mathbf{k}\mathbf{e}\mathbf{r}\mathbf{A}\mathbf{\right)}$ and $\mathit{i}\mathit{m}\mathbf{\left(}{\mathbf{A}}^{\mathbf{T}}\mathbf{\right)}$? Explain.

c. What relationship do you observe betweenrole="math" localid="1660916844921" $\mathit{k}\mathit{e}\mathit{r}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$ and $\mathit{S}$? Explain.

d. Find the unique vector${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$ in the intersection of$\mathit{S}$ and${\mathbf{\left(}\mathbf{k}\mathbf{e}\mathbf{r}\mathbf{A}\mathbf{\right)}}^{\mathbf{鈯�}}$ . Show ${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$on your sketch.

e. What can you say about the length of ${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$compared with the length of all other vectors in $\mathit{S}$?

Recall that a function f(t)fromRtoRis called even if f(鈭�t)=f(t), for all t, and odd if f(鈭�t)= 鈭�f(t), for all t. Show that if f(x)is an odd continuous function and g(x)is an even continuous function, then functions f(x)and g(x)are orthogonal in the space C摆鈭�1,1] with the inner product defined in Example 1.