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For the linear transformations$\mathit{T}$in Exercises$\mathbf{49}$ through$\mathbf{52}$ , do the following:

a. Find thefunction $\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}$definedin Exercise$\mathbf{47}$ and graph itfor $\mathbf{0}\mathbf{鈮�}\mathit{t}\mathbf{鈮�}\frac{\mathbf{蟺}}{\mathbf{2}}$.You may use technology.

b. Find a number$\mathit{c}$ ,with $\mathbf{0}\mathbf{鈮�}\mathit{c}\mathbf{鈮�}\frac{\mathbf{蟺}}{\mathbf{2}}$,suchthat $\mathit{f}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{=}\mathbf{0}$.(In Problem 50, approximate$\mathit{c}$ to three significant digits, using technology.)

c. Find perpendicular unitvectors$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}$and$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}$in${\mathit{鈩�}}^{\mathbf{2}}$suchthat the vectors $\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{\right)}$and$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\right)}$areperpendicular as well. Draw a sketchshowing,$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}$ ,$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}$ ,$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{\right)}$and.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\right)}$

49.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{2}& \mathbf{2}\\ \mathbf{1}& \mathbf{鈭�}\mathbf{4}\end{array}\mathbf{\right)}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$

If possible, compute the matrix products in Exercises 1 through 13, using paper and pencil.

4.$\mathbf{\left[}\begin{array}{cc}1& -1\\ -2& 2\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{cc}7& 5\\ 3& 1\end{array}\mathbf{\right]}$

Find the matrix of linear transformation.

$$\begin{gathered} {\mathbf{y}}_{\mathbf{1}}\mathbf{=}\mathbf{9}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{3}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathbf{x}}_{\mathbf{3}} \\ {\mathbf{y}}_{\mathbf{2}}\mathbf{=}\mathbf{2}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{9}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{3}} \\ {\mathbf{y}}_{\mathbf{3}}\mathbf{=}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{9}{\mathbf{x}}_{\mathbf{2}}\mathbf{-}\mathbf{2}{\mathbf{x}}_{\mathbf{3}} \\ {\mathbf{y}}_{\mathbf{4}}\mathbf{=}\mathbf{5}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{5}{\mathbf{x}}_{\mathbf{3}} \end{gathered}$$

For the linear transformations $\mathit{T}$ in Exercises$\mathbf{49}$ through $\mathbf{52}$do the following:

a. Find the function $\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}$defined in Exercise$\mathbf{47}$ and graph itfor$\mathbf{0}\mathbf{鈮�}\mathit{t}\mathbf{鈮�}\frac{\mathbf{蟺}}{\mathbf{2}}$.You may use technology.

b. Find a number$\mathit{c}$ ,with,$\mathbf{0}\mathbf{鈮�}\mathit{c}\mathbf{鈮�}\frac{\mathbf{蟺}}{\mathbf{2}}$suchthat$\mathit{f}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{=}\mathbf{0}$.(In Problem $\mathbf{50}$50, approximate$\mathit{c}$ to three significant digits, using technology.)

c. Find perpendicular unitvectors$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}$and$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}$in ${\mathit{鈩�}}^{\mathbf{2}}$suchthat the vectors$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{\right)}$ and $\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\right)}$areperpendicular as well. Draw a sketchshowing ,$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}$ ,$\stackrel{鈫�}{v_2}$ ,$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{\right)}$and.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\right)}$

50.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{1}& \mathbf{1}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right)}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$

This Exercise refers to Exercise 49a. Consider the entry$\mathit{k}\mathbf{=}{\mathit{a}}_{\mathbf{11}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{293}$

of the technology matrix . Verify that the entry in the first row and the first column of ${\mathbf{(}{\mathbf{I}}_{\mathbf{n}}\mathbf{鈭�}\mathbf{A}\mathbf{)}}^{\mathbf{-}\mathbf{1}}$is the value of the geometric series$\mathbf{1}\mathbf{+}\mathit{k}\mathbf{+}{\mathit{k}}^{\mathbf{2}}\mathbf{+}\mathbf{鈰�}$ . Interpret this observation in terms of economics.

a. Consider an$\mathit{n}\mathbf{脳}\mathit{m}$ matrix $\mathit{A}$with rank($\mathit{A}$)<$\mathit{n}$. Show that there exists a vector$\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ in${\mathit{鈩�}}^{\mathbf{n}}$ such that the system$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ is inconsistent.Hint: For$\mathit{E}$ =rref($\mathit{A}$), show that there exists a vector$\stackrel{\mathbf{鈫�}}{\mathbf{c}}$in${\mathit{鈩�}}^{\mathbf{n}}$ such that the system$\mathit{E}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{c}}$ is inconsistent; then, 鈥渨ork backword鈥�.

b. Consider an$\mathit{n}\mathbf{脳}\mathit{m}$ matrix $\mathit{A}$with $\mathit{n}$>$\mathit{m}$. Show that there exists a vector $\stackrel{\mathbf{鈫�}}{\mathbf{b}}$in${\mathit{鈩�}}^{\mathbf{n}}$ such that the system$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ is inconsistent.

For each of the mini-Web in Exercise 50 through 52,

1. Find the transition matrix $\mathit{A}$as defined in Exercise 9,
2. Find the equilibrium distribution, and
3. Find the web page(s) with the highest (na茂ve) Page Rank.

Feel free to use technology throughout.

51. $$\begin{gathered} 1鈫�2 \\ 鈫�猡�鈫� \\ 3鈬�4 \end{gathered}$$

Question: For each of the mini-Web in Exercise 50 through 52,

1. Find the transition matrix as defined in Exercise 9,
2. Find the equilibrium distribution, and
3. Find the web page(s) with the highest (na茂ve) Page Rank.

Feel free to use technology throughout.

For $\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}0& 1& 2\\ 0& 2& 4\\ 0& 3& 6\\ 1& 4& 8\end{array}\right]$, find a vector $\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{4}}$such that the system is$\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ inconsistent.

For the linear transformations $\mathit{T}$ inExercises $\mathbf{49}$through,$\mathbf{52}$do the following:

a. Find thefunction $\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}$defined in Exercise$\mathbf{47}$andgraph itfor .$\mathbf{0}\mathbf{鈮�}\mathit{t}\mathbf{鈮�}\frac{\mathbf{蟺}}{\mathbf{2}}$You may use technology.

b. Find a number c,with, $\mathbf{0}\mathbf{鈮�}\mathit{c}\mathbf{鈮�}\frac{\mathbf{蟺}}{\mathbf{2}}$ suchthat.$\mathit{f}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{=}\mathbf{0}$(In Problem50 50, approximate $\mathit{c}$to three significant digits, using technology.)

c. Find perpendicular unitvectors$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}$and$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}$in ${\mathit{鈩�}}^{\mathbf{2}}$suchthat the vectors$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{\right)}$ and $\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\right)}$are perpendicular as well. Draw a sketch showing$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}$ ,$\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}$,$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{\right)}$,and.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\right)}$

52.$\mathit{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{0}& \mathbf{4}\\ \mathbf{5}& \mathbf{鈭�}\mathbf{3}\end{array}\mathbf{\right)}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$