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Chapter 1: Q3E (page 1)

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

\(T:{\mathbb{R}^3} \to {\mathbb{R}^2}\), rotates points (about the origin) through \(\frac{{3\pi }}{2}\)radians (counter-clockwise).

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

Using linear transformation,

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

02

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

Transformation represents the rotation of\(\frac{{3\pi }}{2}\)radian about the origin (counterclockwise).

\(T\left( {{e_1}} \right) = - {e_2}\) and \(T\left( {{e_2}} \right) = {e_1}\).

03

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

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

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

04

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

\({e_1} = \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]\)and\({e_2} = \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]\).

In the equation \(T = Ax\), the matrix \(A\) is the matrix forlinear transformation \(T\).

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

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

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

So, the linear 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

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 \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

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 \({\mathbb{R}^n}\).

In Exercise 23 and 24, make each statement True or False. Justify each answer.

24.

a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).

b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).

c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.

d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.

e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).

In Exercise 19 and 20, choose \(h\) and \(k\) such that the system has

a. no solution

b. unique solution

c. many solutions.

Give separate answers for each part.

19. \(\begin{array}{l}{x_1} + h{x_2} = 2\\4{x_1} + 8{x_2} = k\end{array}\)

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)
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