91Ó°ÊÓ

Find the least-squares solution ${\stackrel{\mathbf{→}}{\mathbf{x}}}^{\mathbf{*}}$of the system $\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$, where $\mathbf{A}\mathbf{=}\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$and$\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{=}\left[\begin{array}{c}3\\ 2\\ 7\end{array}\right]$. Draw a sketch showing the vector $\stackrel{\mathbf{→}}{\mathbf{b}}$, the image of A, the vector$\mathbf{A}{\stackrel{\mathbf{→}}{\mathbf{x}}}^{\mathbf{*}}$, and the vector$\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{-}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$.

Consider the linear space Pof all polynomials, with inner product

$\mathbf{(}\mathbf{f}\mathbf{,}\mathbf{g}\mathbf{)}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{1}}\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathit{g}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathit{d}\mathit{t}$

For three polynomials f, g, and hwe are given the following inner products:

For example,$\left(f,f\right)\mathbf{=}\mathbf{4}$andlocalid="1660796365768" $(g,h)=\left(h,g\right)=3$

a.Find $\left(f,g+h\right)$

b.Findlocalid="1660796420390" $||g+h||$

c.Find ${\mathbf{proj}}_{\mathbf{E}}\mathbf{h}$, where E=span $\left(f,g\right)$. Express your solution as linear combinations of fand g.

d.Find an orthonormal basis of span $\left(f,g,h\right)$.Express the functions in your basis as linear combinations of

f,g, and h.

a.Consider a vector$\stackrel{\mathbf{âŠ\yen }}{\mathbf{v}}$ in ${\mathbf{R}}^{\mathbf{n}}$, and a scalar k. Show that$\mathbf{}\mathbf{\left|}\mathbf{\right|}\mathbf{k}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{}\mathbf{\left|}\mathbf{\right|}\mathbf{=}\mathbf{\left|}\mathbf{k}\mathbf{\right|}\mathbf{.}\mathbf{\left|}\mathbf{\right|}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{\left|}\mathbf{\right|}$

b.Show that if$\stackrel{\mathbf{âŠ\yen }}{\mathbf{v}}$ is a nonzero vector in ${\mathbf{R}}^{\mathbf{n}}$, then

$\stackrel{\mathbf{→}}{\mathbf{u}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\left|}\mathbf{\right|}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{}\mathbf{\left|}\mathbf{\right|}}$is a unit vector.

TRUE OR FALSE

25. $\left[\begin{array}{cc}3& -4\\ 4& 3\end{array}\right]$ is an orthogonal matrix.

Find the least-squares solution$\stackrel{\mathbf{→}}{\mathbf{x}}$ of the system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$, where$\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]$ and$\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{=}\left[\begin{array}{c}5\\ 6\end{array}\right]$. Use only paper and pencil. Draw a sketch.

Find the QR factorization of the matrices$\left[\begin{array}{cc}4& 5\\ 0& 2\\ 0& 14\\ 3& 10\end{array}\right]$.

TRUE OR FALSE

26. If Vis a subspace of ${\mathit{R}}^{\mathbf{n}}$and $\stackrel{\mathbf{→}}{\mathbf{x}}$is a vector in ${\mathit{R}}^{\mathbf{n}}$, then vector $\mathit{p}\mathit{r}\mathit{o}{\mathbf{j}}_{\mathbf{V}}\stackrel{\mathbf{→}}{\mathbf{x}}$must be orthogonal to vector $\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{-}\mathit{p}\mathit{r}\mathit{o}{\mathbf{j}}_{\mathbf{V}}\stackrel{\mathbf{→}}{\mathbf{x}}$.

Find the orthogonal projection of onto the subspace of spanned by and.

Find the Fourier coefficients of the piecewise continuous function

$\mathit{f}\left(t\right)\mathbf{=}\{\begin{array}{l}\begin{array}{ll}0& ift≤0\end{array}\\ \begin{array}{ll}1& ift>0\end{array}\end{array}$

Sketch the graphs of the first few Fourier polynomials.

If A and B are arbitrary$\mathit{n}\mathbf{×}\mathit{n}$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

$\mathit{B}\mathbf{(}\mathbf{A}\mathbf{+}{\mathbf{A}}^{\mathbf{Γ}}\mathbf{)}{\mathit{B}}^{\mathbf{Γ}}$.