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Find the matrix of the linear transformation $\mathit{L}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}{\mathit{A}}^{\mathbf{T}}$ from ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}\mathbf{to}\mathbf{}{\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$towith respect to the basis$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\mathbf{,}\mathbf{}\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\mathbf{,}\mathbf{}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\mathbf{,}\mathbf{}\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

Find the matrix of the linear transformation$\mathit{L}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{A}\mathbf{-}{\mathit{A}}^{\mathbf{T}}$ from ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}\mathbf{to}\mathbf{}{\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$towith respect to the basis,role="math" localid="1660127697622" $\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{-}\mathbf{1}& \mathbf{0}\end{array}\mathbf{\right]}$

The following statements refer to vectors in \({\mathbb{R}^n}\) (or \({\mathbb{R}^m}\)) with the standard inner product. Mark each statement True or False. Justify each answer.

1. The length of every vector is a positive number.
2. A vector v and its negative, \( - {\bf{v}}\), have equal lengths.
3. The distance between u and v is \(\left\| {{\bf{u}} - {\bf{v}}} \right\|\).
4. If \(r\) is any scalar, then \(\left\| {r{\bf{v}}} \right\| = r\left\| {\bf{v}} \right\|\).
5. If two vectors are orthogonal, they are linearly independent.
6. If x is orthogonal, to both u and v, then x must be orthogonal to \({\bf{u}} - {\bf{v}}\).
7. If \({\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2},\) then u and v are orthogonal.
8. If \({\left\| {{\bf{u}} - {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2},\) then u and v are orthogonal.
9. The orthogonal projection of y onto u is a scalar multiple of y.
10. If a vector y coincides with its orthogonal projection onto a subspace \(W\), y is in \(W\).
11. The set of all vectors in \({\mathbb{R}^n}\) orthogonal to one fixed vector is a subspace of \({\mathbb{R}^n}\).
12. If \(W\) is a subspace of \({\mathbb{R}^n}\), then \(W\) and \({W^ \bot }\) have no vectors in common.
13. If \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) is an orthogonal set and if \({c_1},{c_2},\) and \({c_3}\) are scalars, then \(\left\{ {{c_1}{{\bf{v}}_1},{c_2}{{\bf{v}}_2},{c_3}{{\bf{v}}_3}} \right\}\) is an orthogonal set.
14. If a matrix U has orthonormal columns, then \(U{U^T} = I\).
15. A square matrix with orthogonal columns is an orthogonal matrix.
16. If a square matrix has orthonormal columns, then it also has orthonormal rows.
17. If \(W\) is a subspace, then \({\left\| {{{{\mathop{\rm proj}\nolimits} }_W}{\bf{v}}} \right\|^2} + {\left\| {{\bf{v}} - {{{\mathop{\rm proj}\nolimits} }_W}{\bf{v}}} \right\|^2} = {\left\| {\bf{v}} \right\|^2}\).
18. A least-squares solution of \(A{\mathop{\rm x}\nolimits} = b\) is the vector \(A\widehat {\bf{x}}\) in \({\mathop{\rm Col}\nolimits} A\) closest to b, so that \(\left\| {{\mathop{\rm b}\nolimits} - A\widehat {\bf{x}}} \right\| \le \left\| {{\mathop{\rm b}\nolimits} - A{\bf{x}}} \right\|\) for all x.
19. The normal equations for a least-squares solution of \(A{\mathop{\rm x}\nolimits} = b\) are given by \(\widehat {\mathop{\rm x}\nolimits} = {\left( {{A^T}A} \right)^{ - 1}}{A^T}{\mathop{\rm b}\nolimits} \).

Consider the matrix $\mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& 1& -1\\ 3& 2& -5\\ 2& 2& 0\end{array}\right]$ with LDU-factorizationrole="math" localid="1660129574693" $\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{0}& \mathbf{0}\\ \mathbf{3}& \mathbf{1}& \mathbf{0}\\ \mathbf{2}& \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{-}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{2}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{1}& \mathbf{-}\mathbf{1}\\ \mathbf{0}& \mathbf{1}& \mathbf{2}\\ \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}$. Find the LDU-factorization for${\mathit{A}}^{\mathbf{T}}$.

Consider a symmetric invertible n脳nmatrix Awhich admits an LDU-factorization A=LDU. See Exercises 90, 93, and 94 of Section 2.4. Recall that this factorization is unique. See Exercise 2.4.94. Show that

$\mathit{U}\mathbf{=}{\mathit{L}}^{\mathbf{T}}$ (This is sometimes called the$\mathit{L}\mathit{D}{\mathit{L}}^{\mathbf{T}}$ - factorizationof a symmetric matrix A.)

This exercise shows one way to define the quaternions,discovered in 1843 by the Irish mathematician Sir W.R. Hamilton (1805-1865).Consider the set H of all $\mathbf{4}\mathbf{脳}\mathbf{4}$matrices M of the form

$\mathit{M}\mathbf{=}\left[\begin{array}{cccc}p& 鈭�q& 鈭�r& 鈭�s\\ q& p& s& 鈭�r\\ r& 鈭�s& p& q\\ s& r& 鈭�q& p\end{array}\right]$

where p,q,r,s are arbitrary real numbers.We can write M more sufficiently in partitioned form as

$\mathit{M}\mathbf{=}\left(\begin{array}{cc}A& 鈭�B^T\\ B& A^T\end{array}\right)$

where A and B are rotation-scaling matrices.

a.Show that H is closed under addition:If M and N are in H then so is$\mathit{M}\mathbf{+}\mathit{N}$

M+Nb.Show that H is closed under scalar multiplication .If M is in H and K is an arbitrary scalar then kM is in H.

c.Parts (a) and (b) Show that H is a subspace of the linear space ${\mathit{R}}^{\mathbf{4}\mathbf{脳}\mathbf{4}}$ .Find a basis of H and thus determine the dimension of H.

d.Show that H is closed under multiplication If M and N are in H then so is MN.

e.Show that if M is in H,then so is ${\mathit{M}}^{\mathbf{T}}$.

f.For a matrix M in H compute ${\mathit{M}}^{\mathbf{T}}\mathit{M}$.

g.Which matrices M in H are invertible.If a matrix M in H is invertible is${\mathit{M}}^{\mathbf{鈭�}\mathbf{1}}$ necessarily in H as well?

h. If M and N are in H,does the equation$\mathit{M}\mathit{N}\mathbf{=}\mathit{N}\mathit{M}$always hold?

Consider a subspace V of ${\mathbf{鈻�}}^{\mathbf{n}}$with a basis ${\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{m}}$; suppose we wish to find a formula for the orthogonal projection onto V. Using the methods we have developed thus far, we can proceed in two steps: We use the Gram-Schmidt process to construct an orthonormal basis${\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{u}}}_{\mathbf{m}}$ of V, and then we use Theorem 5.3.10: The matrix of the orthogonal projection QQ^T, where $\mathit{Q}\mathbf{=}\left[{\stackrel{鈬赌}{u}}_1...{\stackrel{鈬赌}{u}}_m\right]$. In this exercise we will see how we write the matrix of the projection directly in terms of the basis${\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{m}}$ and the matrix $\mathit{A}\mathbf{=}\left[{\stackrel{鈬赌}{v}}_1...{\stackrel{鈬赌}{v}}_m\right]$. (This issue will be discussed more thoroughly in Section 5.4: see theorem 5.4.7.)

Since $\mathit{p}\mathit{r}\mathit{o}{\mathit{j}}_{\mathbf{v}}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$ is in V, we can write $\mathit{p}\mathit{r}\mathit{o}{\mathit{j}}_{\mathbf{v}}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{=}\mathit{c}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathit{c}}_{\mathbf{m}}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{m}}$for some scalars ${\mathit{c}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{c}}_{\mathbf{m}}$yet to be determined. Now $\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{-}\mathit{p}\mathit{r}\mathit{o}{\mathit{j}}_{\mathbf{v}}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{-}{\mathit{c}}_{\mathbf{1}}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{-}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{-}{\mathit{c}}_{\mathbf{m}}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{m}}$ is orthogonal to V, meaning that role="math" localid="1660623171035" ${\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{i}}\mathbf{路}\mathbf{(}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{-}{\mathbf{c}}_{\mathbf{1}}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{-}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{-}{\mathbf{c}}_{\mathbf{m}}{\stackrel{\mathbf{鈬赌}}{\mathbf{v}}}_{\mathbf{m}}\mathbf{)}\mathbf{=}\mathbf{0}$for $\mathit{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{m}$.

In Exercises 5-14, the space is \(C\left[ {0,2\pi } \right]\) with inner product (6).

10. Find the third-order Fourier approximation to square wave function \(f\left( t \right) = 1{\rm{ for }}0 \le t < \pi \) and \(f\left( t \right) = - 1{\rm{ for }}\pi \le t < 2\pi \).

In Exercises 5-14, the space is \(C\left[ {0,2\pi } \right]\) with inner product (6).

11. Find the third-order Fourier approximation to \({\sin ^2}t\), without performing any integration calculations.

In Exercises 5-14, the space is \(C\left[ {0,2\pi } \right]\) with inner product (6).

12. Find the third-order Fourier approximation to \({\cos ^3}t\), without performing any integration calculations.