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Chapter 1: Q13E (page 1)

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation such that \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) are the vectors shown in the figure. Using the figure, sketch the vector \(T\left( {2,1} \right)\).

Short Answer

Expert verified

Using linear transformation \(\left( {2,1} \right)\) can be written as follows:

\(\left( {2,1} \right) = 2{e_1} + {e_2}\)

Step by step solution

01

Write \(T\left( {2,1} \right)\) in the form of \({e_1}\) and \({e_2}\)

Using linear transformation \(\left( {2,1} \right)\) can be written as follows:

\(\left( {2,1} \right) = 2{e_1} + {e_2}\)

02

Using linearity of \(T\) find image of \(\left( {2,1} \right)\)

\(\begin{aligned} T\left( {2,1} \right) &= T\left( {2{e_1}} \right) + T\left( {{e_2}} \right)\\ &= 2T\left( {{e_1}} \right) + T\left( {{e_2}} \right)\end{aligned}\)

03

Locate \(2T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) in the graph

In the given graph, locate \(2T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) to form a parallelogram as shown below:

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Most popular questions from this chapter

Question: Determine whether the statements that follow are true or false, and justify your answer.

16: There exists a 2x2 matrix such that

A[11]=[12]andA[22]=[21].

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

16. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)

Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure.

  1. \(\begin{aligned}{c}{x_1} + 5{x_2} = 7\\ - 2{x_1} - 7{x_2} = - 5\end{aligned}\)

Find the general solutions of the systems whose augmented matrices are given as

14. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&{ - 6}&0&{ - 5}\\0&1&{ - 6}&{ - 3}&0&2\\0&0&0&0&1&0\\0&0&0&0&0&0\end{array}} \right]\).

Consider a dynamical system x→(t+1)=Ax→(t)with two components. The accompanying sketch shows the initial state vector x→0and two eigenvectors υ1→  and  υ2→of A (with eigen values λ1→andλ2→ respectively). For the given values of λ1→andλ2→, draw a rough trajectory. Consider the future and the past of the system.

λ1→=1.2,λ2→=1.1
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