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Chapter 1: Q5E (page 1)

In Exercise 1-10, assume that \(T\) is a linear transformation. Find the standard matrix of \(T\).

\(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\), is a vertical shear transformation that maps \({e_1}\) into \({e_1} - 2{e_2}\) but leaves the vector \({e_2}\) unchanged.

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&0\\{ - 2}&1\end{array}} \right]\)

Step by step solution

01

Find the value of \(T\) using linear transformation

Usinglinear transformation,

\(\begin{aligned} T &= T\left( {{x_1}{e_1} + {x_2}{e_2}} \right)\\ &= {x_1}T\left( {{e_1}} \right) + {x_2}T\left( {{e_2}} \right)\\ &= \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}\end{array}} \right]x\end{aligned}\)

02

Find the transformation for \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\)

Here,

\(\begin{aligned} T\left( {{e_1}} \right) &= {e_1} - 2{e_2}\\ &= \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right] - 2\left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\end{array}} \right]\end{aligned}\)

And

\(\begin{aligned} T\left( {{e_2}} \right) &= {e_2}\\ &= \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]\end{aligned}\)

03

Find the transformation matrix for \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\)

By the equation \(T = \left[ {\begin{array}{*{20}{c}}{T\left( {{e_1}} \right)}&{T\left( {{e_2}} \right)}\end{array}} \right]x\),

\(T = \left[ {\begin{array}{*{20}{c}}1&0\\{ - 2}&1\end{array}} \right]x\).

04

Find the standard matrix \(T\) for linear transformation

By the equation \(T = \left[ {\begin{array}{*{20}{c}}1&0\\{ - 2}&1\end{array}} \right]x\), the matrix \(A = \left[ {\begin{array}{*{20}{c}}1&0\\{ - 2}&1\end{array}} \right]\).

So, the linear transformation matrix is \(\left[ {\begin{array}{*{20}{c}}1&0\\{ - 2}&1\end{array}} \right]\).

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Most popular questions from this chapter

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a de铿乶ition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

23.

a. Every elementary row operation is reversible.

b. A \(5 \times 6\)matrix has six rows.

c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.

d. Two fundamental questions about a linear system involve existence and uniqueness.

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.

a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).

b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).

Find an equation involving \(g,\,h,\)and \(k\) that makes this augmented matrix correspond to a consistent system:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]\)

Consider the dynamical system x鈫�(t+1)=[1.100鈪�]X鈫�(t).

Sketch a phase portrait of this system for the given values of 位:

位=1

Use Theorem 7 in section 1.7 to explain why the columns of the matrix Aare linearly independent.

\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)
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