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Chapter 1: Q4E (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\( - \frac{\pi }{4}\)radians (clockwise).[Hint:\(T\left( {{e_1}} \right) = \left[ {\frac{1}{{\sqrt 2 }}, - \frac{1}{{\sqrt 2 }}} \right]\)]

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\)

Step by step solution

01

Find the value of \(T\) using linear transformation

Usinglinear 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 \( - \frac{\pi }{4}\) radian about the origin (counterclockwise).

\(T\left( {{e_1}} \right) = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}\end{array}} \right]\)

And

\(T\left( {{e_2}} \right) = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\)

03

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

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

\(T = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]x\).

04

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

By the equation \(T = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]x\), the matrix \(A = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\).

So, the linear transformation matrix is \(\left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\{ - \frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\end{array}} \right]\).

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Most popular questions from this chapter

In Exercises 32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

32. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&{ - 3}&9&5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&0&0&{ - 1}\end{array}} \right]\)

Find the polynomial of degree 2[a polynomial of the form f(t)=a+bt+ct2] whose graph goes through the points localid="1659342678677" (1,-1),(2,3)and(3,13).Sketch the graph of the polynomial.

Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.

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 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 definition 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.

23.

a. Every elementary row operation is reversible.

b. A \(5 \times 6\)matrix has six rows.

c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.

d. Two fundamental questions about a linear system involve existence and uniqueness.
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