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Chapter 1: Q6E (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 horizontal shear transformation that leaves \({e_1}\) unchanged and \({e_2}\) into \({e_2} + 3{e_1}\).

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&3\\0&1\end{array}} \right]\)

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,

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

And

\(\begin{aligned} T\left( {{e_2}} \right) &= {e_2} + 3{e_1}\\ &= \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right] + 3\left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}3\\1\end{array}} \right]\end{aligned}\)

03

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

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

\(T = \left[ {\begin{array}{*{20}{c}}1&3\\0&1\end{array}} \right]x\).

04

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

By the equation \(T = \left[ {\begin{array}{*{20}{c}}1&3\\0&1\end{array}} \right]x\), the matrix \(A = \left[ {\begin{array}{*{20}{c}}1&3\\0&1\end{array}} \right]\).

So, thelinear transformation matrix is \(\left[ {\begin{array}{*{20}{c}}1&3\\0&1\end{array}} \right]\).

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Most popular questions from this chapter

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.a

Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a plane in \({\mathbb{R}^3}\).

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.

24.

a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

b. Two matrices are row equivalent if they have the same number of rows.

c. An inconsistent system has more than one solution.

d. Two linear systems are equivalent if they have the same solution set.

In Exercise 1, compute \(u + v\) and \(u - 2v\).

  1. \(u = \left[ {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}{ - 3}\\{ - 1}\end{array}} \right]\).

In Exercises 13 and 14, determine if \({\mathop{\rm b}\nolimits} \) is a linear combination of the vectors formed from the columns of the matrix \(A\).

14. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 6}\\0&3&7\\1&{ - 2}&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}{11}\\{ - 5}\\9\end{array}} \right]\)
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