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In Exercises19through 24 , find the matrix Bof the linear transformation $\mathit{T}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$ with respect to the basis $\mathfrak{I}\mathbf{=}\left(\stackrel{鈬赌}{v_1},\stackrel{鈬赌}{v_2}\right)$. For practice, solve each problem in three ways: (a) Use the formula $\mathit{B}\mathbf{=}{\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}$ , (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct 鈥渃olumn by column.鈥�

localid="1664194720187" $\mathit{A}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\mathbf{\right)}\mathbf{;}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{1}\end{array}\mathbf{\right]}\mathbf{;}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{-}\mathbf{1}\end{array}\mathbf{\right]}$

In Exercises 9through 24, find the matrix Bof the linear transformation $\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$ with respect to the basis $\mathfrak{I}\mathbf{=}\left(\stackrel{鈬赌}{v_1},\stackrel{鈬赌}{v_2}\right)$. For practice, solve each problem in three ways: (a) Use the formula $\mathit{B}\mathbf{=}{\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}$, (b) use a commutative diagram (as in Examples 3 and 4 ), and (c)construct 鈥渃olumn by column.鈥�

role="math" localid="1664342971664" $\mathit{A}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{3}& \mathbf{6}\end{array}\mathbf{\right)}\mathbf{;}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{3}\end{array}\mathbf{\right]}\mathbf{;}\stackrel{\mathbf{鈬赌}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{-}\mathbf{2}\\ \mathbf{1}\end{array}\mathbf{\right]}$

Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points $(a_i,b_j)$are given to you, and your job is to connect the dots in a reasonably smooth way. Let ${\mathbf{a}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\mathbf{>}{\mathbf{a}}_{\mathbf{i}\mathbf{}}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}\mathbf{-}\mathbf{1}$.

One method often employed in such design problems is the technique of cubic splines. We choose ${\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$, a polynomial of degree $\mathbf{鈮�}\mathbf{3}$, to define the shape of the ride between $(a_{i-1},b_{i-1})$and $\mathbf{(}{\mathbf{a}}_{\mathbf{i}}\mathbf{,}{\mathbf{b}}_{\mathbf{j}}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}$.

Obviously, it is required that ${\mathbf{f}}_{\mathbf{i}}\left(a_i\right)\mathbf{=}{\mathbf{b}}_{\mathbf{i}}$and ${\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}\mathbf{-}\mathbf{1}}\mathbf{\right)}\mathbf{=}{\mathbf{b}}_{\mathbf{i}\mathbf{-}\mathbf{1}}\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}$. To guarantee a smooth ride at the points $\left(a_i,b_i\right)$, we want the first and second derivatives of ${\mathbf{f}}_{\mathbf{i}}$and ${\mathbf{f}}_{\mathbf{i}\mathbf{+}\mathbf{1}}$to agree at these points:

$\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}}\mathbf{\right)}\mathbf{=}\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}}\mathbf{\right)}$and

$\mathbf{f}\mathbf{\text{'}}{\mathbf{\text{'}}}_{\mathbf{i}}\left(a_i\right)\mathbf{=}\mathbf{f}\mathbf{\text{'}}{\mathbf{\text{'}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\left(a_i\right)\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{i}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{.}$

Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required that

$\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{1}}\left(a_0\right)\mathbf{=}\mathbf{f}{\mathbf{\text{'}}}_{\mathbf{n}}\left(a_n\right)\mathbf{=}\mathbf{0}$

Show that satisfying all these conditions amounts to solving a system of linear equations. How many variables are in this system? How many equations? (Note: It can be shown that this system has a unique solution.)

We define the vectors

$\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{1}}}\mathbf{=}\left[\left.\begin{array}{c}1\\ 0\\ 0\end{array}\right],\stackrel{鈫�}{e_2}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\stackrel{鈫�}{e_3}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right.\right]$

in${\mathit{R}}^{\mathbf{3}}$.

a. For role="math" localid="1659342928825" $\mathbf{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{a}& \mathbf{b}& \mathbf{c}\\ \mathbf{d}& \mathbf{e}& \mathbf{f}\\ \mathbf{g}& \mathbf{h}& \mathbf{k}\end{array}\mathbf{\right]}$

compute role="math" localid="1659343034980" $\mathit{A}\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{1}}}\mathbf{,}\mathit{A}\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{2}}}$and role="math" localid="1659343045854" $\mathit{A}\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{3}}}$.

b. If B is an role="math" localid="1659343084344" $\mathbf{n}\mathbf{脳}\mathbf{3}$matrix with columns $\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}$and $\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}$, what are role="math" localid="1659343268769" $\mathit{B}\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{1}}}\mathbf{,}\mathit{B}\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{2}}}$and $\mathit{B}\stackrel{\mathbf{鈫�}}{{\mathbf{e}}_{\mathbf{3}}}$?

Exercise 33 illustrates how you can use the powers of a matrix to find its dominant eigenvalue (i.e., the eigenvalue with maximal modulus), at least when this eigenvalue is real. But what about the other eigenvalues?

a. Consider a $\mathit{n}\mathit{x}\mathit{n}$matrix A with n distinct complex eigenvalues${\mathit{位}}_{\mathbf{1}}\mathbf{,}{\mathit{位}}_{\mathbf{2}}\mathbf{,}\mathbf{鈥�}\mathbf{,}{\mathit{位}}_{\mathbf{n}}\mathbf{,}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{}{\mathit{位}}_{\mathbf{1}}$1 is real. Suppose you have a good (real) approximation $\mathit{位}$ of ${\mathit{位}}_{\mathbf{1}}$ (good in that $\mathbf{|}\mathbf{位}\mathbf{-}{\mathbf{位}}_{\mathbf{1}}\mathbf{|}\mathbf{<}\mathbf{|}\mathbf{位}\mathbf{-}{\mathbf{位}}_{\mathbf{i}}\mathbf{|}$for $\mathit{i}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{鈥�}\mathbf{,}\mathit{n}\mathbf{)}$Consider the matrix $\mathit{A}\mathbf{-}\mathit{位}{\mathit{I}}_{\mathbf{n}}$. What are its eigenvalues? Which has the smallest modulus? Now consider the matrix ${\mathbf{(}\mathbf{A}\mathbf{-}\mathbf{位}{\mathbf{I}}_{\mathbf{n}}\mathbf{)}}^{\mathbf{-}\mathbf{1}}$What are its eigenvalues? Which has the largest modulus? What is the relationship between the eigenvectors of A and those of ${\mathbf{(}\mathbf{A}\mathbf{-}\mathbf{位}{\mathbf{I}}_{\mathbf{n}}\mathbf{)}}^{\mathbf{-}\mathbf{1}}$? Consider higher and higher powers of .${\mathbf{(}\mathbf{A}\mathbf{-}\mathbf{位}{\mathbf{I}}_{\mathbf{n}}\mathbf{)}}^{\mathbf{-}\mathbf{1}}$How does this help you to find an eigenvector of A with eigenvalue${\mathit{位}}_{\mathbf{1}}$, and${\mathit{位}}_{\mathbf{1}}$itself? Use the results of Exercise 33.

b. As an example of part a, consider the matrix $\mathit{A}\mathbf{=}\mathbf{[}\begin{array}{l}\mathbf{1}\\ \mathbf{4}\\ \mathbf{7}\end{array}$$\begin{array}{c}\mathbf{2}\\ \mathbf{5}\\ \mathbf{8}\end{array}$$\begin{array}{r}\mathbf{3}\\ \mathbf{6}\\ \mathbf{10}\end{array}\mathbf{]}$$\begin{array}{c}\mathbf{2}\\ \mathbf{5}\\ \mathbf{8}\end{array}$

We wish to find the eigenvectors and eigenvalues of without using the corresponding commands on the computer (which is, after all, a 鈥渂lack box鈥�). First, we find approximations for the eigenvalues by graphing the characteristic polynomial (use technology). Approximate the three real eigenvalues of to the nearest integer. One of the three eigenvalues of is negative. Find a good approximation for this eigenvalue and a corresponding eigenvector by using the procedure outlined in part a. Youare not asked to do the same for the two other eigenvalues.

Find the polynomial of degree $\mathbf{鈮�}\mathbf{2}$[a polynomial of the form $\mathbf{f}\left(t\right)\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{bt}\mathbf{+}{\mathbf{ct}}^{\mathbf{2}}$] whose graph goes through the points $\left(1,1\right)\mathbf{,}\left(2,0\right)$, such that ${\mathbf{鈭�}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{f}\left(t\right)\mathbf{d}\mathbf{t}\mathbf{=}\mathbf{-}\mathbf{1}$

The function $\mathbf{q}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\stackrel{\mathbf{鈫�}\mathbf{T}}{\mathbf{x}}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\stackrel{\mathbf{鈫�}}{\mathbf{x}}$is a quadratic form.

Consider the subspace Wof ${\mathbf{R}}^{\mathbf{4}}$spanned by the vectors${\stackrel{\mathbf{鈬赌}}{\mathbf{V}}}_{\mathbf{1}}\mathbf{=}\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\mathbf{}\mathbf{and}\mathbf{}{\stackrel{\mathbf{鈬赌}}{\mathbf{V}}}_{\mathbf{1}}\mathbf{=}\left[\begin{array}{c}1\\ 9\\ -5\\ 3\end{array}\right]$

and Find the matrix of the orthogonal projection onto W.

If ${\mathit{A}}^{\mathbf{2}}$ is invertible, then matrix A itself must be invertible.

Find the polynomial f(t) of degree 3 such that $\mathit{f}\left(1\right)\mathbf{=}\left(1\right)\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathbf{,}\mathit{f}\mathbf{\text{'}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}$and $\mathit{f}\mathbf{\text{'}}\left(2\right)\mathbf{=}\mathbf{9}$, where $\mathit{f}\mathbf{\text{'}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ is the derivative of $\mathit{f}\left(t\right)$. Graph this polynomial.