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Chapter 1: Q8E (page 1)

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

\(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\), firsts reflects points through the horizontal \({x_1}\)-axis and then reflects point through the line \({x_2} = {x_1}\).

Short Answer

Expert verified

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

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

Finding the transformation \(T\) for \({e_1}\)

For \({e_1}\), when it is reflected through the horizontal \({x_1}\)-axis, then

\({e_1} \to {e_1}\).

When it is reflected through the line \({x_2} = {x_1}\),

\({e_1} \to {e_2}\).

03

Finding the transformation \(T\) for \({e_2}\)

For \({e_2}\), when it is reflected through the horizontal \({x_1}\)-axis, then

\({e_2} \to - {e_2}\).

When it is reflected through the line \({x_2} = {x_1}\),

\( - {e_2} \to - {e_1}\).

04

Find the transformation matrix

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

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

05

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

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

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

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Most popular questions from this chapter

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

5. \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?

7.Vectors a, b, c, and d

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)

Explain why a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) in \({\mathbb{R}^5}\) must be linearly independent when \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly independent and \({{\mathop{\rm v}\nolimits} _4}\) is not in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).
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