91Ó°ÊÓ

TRUE OR FALSE

If$\mathbf{\left|}\mathbf{\right|}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left|}\mathbf{\right|}\mathbf{â\copyright ½}\mathbf{\left|}\mathbf{\right|}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left|}\mathbf{\right|}$for $\stackrel{\mathbf{→}}{\mathbf{x}}$all in ${\mathit{R}}^{\mathbf{n}}$, then Amust represent the orthogonal projection onto a subspace Vof ${\mathit{R}}^{\mathbf{n}}$.

Find an orthonormal basis of the plane ${\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{3}}\mathbf{=}\mathbf{0}$.

Consider two vectors ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}$and localid="1659542347261" ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$in ${\mathit{R}}^{\mathbf{n}}$. Form the matrix

$\mathit{G}\mathbf{=}\left[\begin{array}{cc}{\stackrel{→}{v}}_1.{\stackrel{→}{v}}_1& {\stackrel{→}{v}}_1.{\stackrel{→}{v}}_2\\ {\stackrel{→}{v}}_2.{\stackrel{→}{v}}_1& {\stackrel{→}{v}}_2.{\stackrel{→}{v}}_2\end{array}\right]$

For which choices of localid="1659542539965" ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}$and ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$is the matrix Ginvertible?

Fit a quadratic polynomial to the data points (1,0) , (2,0) , (3,0) using least square. Sketch the solution.

TRUE OR FALSE

If Ais an invertible matrix such that ${\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathit{A}$, then Amust be orthogonal.

Among all the vectors in ${\mathbf{R}}^{\mathbf{n}}$whose components add up to 1, find the vector of minimal length. In the case$\mathbf{n}\mathbf{=}\mathbf{2}$ , explain your solution geometrically.

Find an orthonormal basis of the kernel of the matrix $\mathbf{A}\mathbf{=}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& -1& 1\end{array}\right]$.

Find all orthogonal 2 × 2 matrices.

a. Let $\mathbf{w}\left(t\right)$be a positive-valued function in $\mathbf{C}\mathbf{[}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{]}$, where $\mathit{b}\mathbf{>}\mathit{a}$. Verify that the rule $\left(f,g\right)\mathbf{=}{\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathbf{w}\left(t\right)\mathbf{f}\left(t\right)\mathbf{g}\left(t\right)\mathbf{dt}$defines an inner product on $\mathbf{C}\left[a,b\right]$.

b. If we chose the weight function $\mathbf{w}\left(t\right)$so that ${\mathbf{∫}}_{\mathbf{a}}^{\mathbf{b}}\mathbf{w}\left(t\right)\mathbf{dt}\mathbf{=}\mathbf{1}$, what is the norm of the constant function $\mathbf{f}\left(t\right)\mathbf{=}\mathbf{1}$in this inner product space?

- Find the trigonometric function of the form $\mathit{f}\left(t\right)\mathbf{=}{\mathit{c}}_{\mathbf{0}}\mathbf{+}{\mathit{c}}_{\mathbf{1}}\mathbf{sin}\left(t\right)\mathbf{+}{\mathit{c}}_{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\left(t\right)$fits the data points$\left(0,0\right)\mathbf{,}\left(1,1\right)\mathbf{,}\left(2,2\right)\mathbf{,}\left(3,3\right)$ using least square. Sketch the solution together with the function