91Ó°ÊÓ

Question: In Exercises 1 and 2, you may assume that\(\left\{ {{{\bf{u}}_{\bf{1}}},...,{{\bf{u}}_{\bf{4}}}} \right\}\)is an orthogonal basis for\({\mathbb{R}^{\bf{4}}}\).

1.\({{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\\{ - {\bf{1}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{5}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{3}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{4}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{ - {\bf{1}}}\\{\bf{1}}\end{aligned}} \right]\),\({\bf{x}} = \left[ {\begin{aligned}{*{20}{c}}{{\bf{10}}}\\{ - {\bf{8}}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right]\)

Write x as the sum of two vectors, one in\({\bf{Span}}\left\{ {{{\bf{u}}_1},{{\bf{u}}_2},{{\bf{u}}_3}} \right\}\)and the other in\({\bf{Span}}\left\{ {{{\bf{u}}_{\bf{4}}}} \right\}\).

For which value(s) of the constant k are the vectors

$\stackrel{\mathbf{→}}{\mathbf{u}}\mathbf{=}\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{v}}\left[\begin{array}{c}1\\ k\\ 1\end{array}\right]$and perpendicular?

Consider a consistent system $\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$.

(a) Show that this system has a solution ${\stackrel{\mathbf{→}}{\mathbf{x}}}_{\mathbf{0}}$ in ${\left(kerA\right)}^{\mathbf{âŠ\yen }}$ .

(b) Show that the system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$ has only one solution in ${\left(kerA\right)}^{\mathbf{âŠ\yen }}$ .

(c) If${\stackrel{\mathbf{→}}{\mathbf{x}}}_{\mathbf{0}}$ is the solution in ${\mathbf{\left(}\mathbf{kerA}\mathbf{\right)}}^{\mathbf{âŠ\yen }}$ androle="math" localid="1660124695419" ${\stackrel{\mathbf{→}}{\mathbf{x}}}_1$is another solution of the system $\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$ , show that$||{\stackrel{→}{x}}_0||\mathbf{<}||{\stackrel{→}{x}}_1||$ . The vector${\stackrel{\mathbf{→}}{\mathbf{x}}}_{\mathbf{0}}$ is called the minimal solution of the linear system $\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$ .

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

10.$\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}6\\ 4\\ 6\\ 4\end{array}\right]$

If is a symmetric matrix, then must be symmetric as well.

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

Consider the space${\mathit{P}}_{\mathbf{2}}$ with inner product

$<f,g>\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{∫}}_{\mathbf{-}\mathbf{1}}^{\mathbf{1}}\mathit{f}\left(t\right)\mathit{g}\left(t\right)\mathit{d}\mathit{t}$

Find an orthonormalbasis of the space of all functions in that are orthogonal to f(t)=t.

The angle between two nonzero elementsvandwof an inner product space is defined as

$\mathbf{∡}\left(v,w\right)\mathbf{=}\mathit{a}\mathit{r}\mathit{c}\mathit{c}\mathit{o}\mathit{s}\frac{<v,w>}{||v||\mathbf{.}||w||}$

In the space $\left[-\mathrm{Ï€},\mathrm{Ï€}\right]$with inner product

$<f,g>\mathbf{=}\frac{\mathbf{1}}{\mathbf{π}}\underset{\mathbf{-}\mathbf{π}}{\overset{\mathbf{π}}{\mathbf{∫}}}\mathit{f}\left(t\right)\mathit{g}\left(t\right)\mathit{d}\mathit{t}$

find the angle between $\mathit{f}\left(t\right)\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\left(t\right)$and $\mathit{g}\left(t\right)\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\left(t+\mathrm{δ}\right)$where $\mathbf{0}\mathbf{≤}\mathit{δ}\mathbf{≤}\mathbf{π}$. Hint: Use the formula $\mathit{c}\mathit{o}\mathit{s}\left(t+\mathrm{δ}\right)\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{δ}\mathbf{\right)}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathit{δ}\mathbf{\right)}$.

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

11.$\mathbf{\left[}\begin{array}{c}\mathbf{2}\\ \mathbf{3}\\ \mathbf{0}\\ \mathbf{6}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{c}\mathbf{4}\\ \mathbf{4}\\ \mathbf{2}\\ \mathbf{13}\end{array}\mathbf{\right]}$

TRUE OR FALSE?If $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{x}}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{y}}$are two vectors in${\mathit{R}}^{\mathbf{n}}$, then the equation role="math" localid="1659506190737" $\mathbf{âˆ\yen }\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{y}}{\mathbf{âˆ\yen }}^{\mathbf{2}}\mathbf{=}\mathbf{âˆ\yen }\stackrel{\mathbf{→}}{\mathbf{x}}{\mathbf{âˆ\yen }}^{\mathbf{2}}\mathbf{+}\mathbf{âˆ\yen }\stackrel{\mathbf{→}}{\mathbf{y}}{\mathbf{âˆ\yen }}^{\mathbf{2}}$must hold.