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Chapter 1: Q17E (page 1)

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be a linear transformation that maps \(u = \left[ {\begin{array}{*{20}{c}}5\\2\end{array}} \right]\) into \(\left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\) and maps \(v = \left[ {\begin{array}{*{20}{c}}1\\3\end{array}} \right]\) into \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\). Use the fact that \(T\) is linear to find the images under \(T\) of \(3u\), \(2v\) and \(3u + 2v\).

Short Answer

Expert verified

\(T\left( {3u} \right) = \left[ {\begin{array}{*{20}{c}}6\\3\end{array}} \right]\), \(T\left( {2v} \right) = \left[ {\begin{array}{*{20}{c}}{ - 2}\\6\end{array}} \right]\), and \(T\left( {3u + 2v} \right) = \left[ {\begin{array}{*{20}{c}}4\\9\end{array}} \right]\)

Step by step solution

01

Finding the transformed coordinates

The transformed rectangular coordinate of \(u\) is:

\(T\left( u \right) = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\)

The transformed rectangular coordinate of \(v\) is:

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

02

Finding the image of rectangular coordinates

Calculate the value of \(T\left( {3u} \right)\) as follows:

\(\begin{aligned}{c}T\left( {3u} \right) &= 3T\left( u \right)\\ &= 3\left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}6\\3\end{array}} \right]\end{aligned}\)

03

Finding the image of the rectangular coordinates

Calculate the value of \(T\left( {2v} \right)\) as follows:

\(\begin{aligned}T\left( {2v} \right) &= 2T\left( v \right)\\ &= 2\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 2}\\6\end{array}} \right]\end{aligned}\)

04

Finding the image of the rectangular coordinates

Calculate the value of \(T\left( {3u + 2v} \right)\) as follows:

\(\begin{aligned} T\left( {3u + 2v} \right) &= T\left( {3u} \right) + T\left( {2v} \right)\\ &= \left[ {\begin{array}{*{20}{c}}6\\3\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ - 2}\\6\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}4\\9\end{array}} \right]\end{aligned}\)

So, the values of the images are \(T\left( {3u} \right) = \left[ {\begin{array}{*{20}{c}}6\\3\end{array}} \right]\), \(T\left( {2v} \right) = \left[ {\begin{array}{*{20}{c}}{ - 2}\\6\end{array}} \right],\)and \(T\left( {3u + 2v} \right) = \left[ {\begin{array}{*{20}{c}}4\\9\end{array}} \right]\).

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Most popular questions from this chapter

Construct a \(3 \times 3\) matrix\(A\), with nonzero entries, and a vector \(b\) in \({\mathbb{R}^3}\) such that \(b\) is not in the set spanned by the columns of\(A\).

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

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

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

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 \({\mathbb{R}^{\bf{3}}}\).

Show that if ABis invertible, so is B.
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