Q52E For the linear transformations T... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q52E (page 74)

For the linear transformations $T$ inExercises $49$ through, $52$ do the following:
a. Find thefunction $f (t)$ defined in Exercise $47$ andgraph itfor . $0 鈮� t 鈮� \frac{蟺}{2}$ You may use technology.
b. Find a number c,with, $0 鈮� c 鈮� \frac{蟺}{2}$ suchthat. $f (c) = 0$ (In Problem50 50, approximate $c$ to three significant digits, using technology.)
c. Find perpendicular unitvectors $\overset{鈫�}{v_{1}}$ and $\overset{鈫�}{v_{2}}$ in ${鈩�}^{2}$ suchthat the vectors $T (\overset{鈫�}{v_{1}})$ and $T (\overset{鈫�}{v_{2}})$ are perpendicular as well. Draw a sketch showing $\overset{鈫�}{v_{1}}$ , $\overset{鈫�}{v_{2}}$ , $T (\overset{鈫�}{v_{1}})$ ,and. $T (\overset{鈫�}{v_{2}})$
52. $T (\overset{鈫�}{x}) = (\begin{matrix} 0 & 4 \\ 5 & 鈭� 3 \end{matrix}) \overset{鈫�}{x}$

Short Answer

Expert verified

a. The function is, $f (t) = 15 (2 \sin^{2} t 鈭� 1)$ .

b. The number is,. $c = \frac{蟺}{4}$

c. The perpendicular unit vectors are, $\overset{鈫�}{v_{1}} = \frac{1}{\sqrt{2}} (\begin{array}{l} 1 \\ 1 \end{array}), \overset{鈫�}{v_{2}} = \frac{1}{\sqrt{2}} (\begin{array}{l} 鈭� 1 \\ 1 \end{array})$

Step by step solution

Compute the function.

Consider the matrix.

$T (\overset{鈫�}{x}) = (\begin{matrix} 0 & 4 \\ 5 & 鈭� 3 \end{matrix}) \overset{鈫�}{x}$

The function is,

$\begin{array}{l} f (t) = (\begin{matrix} 0 & 4 \\ 5 & 鈭� 3 \end{matrix}) (\begin{array}{l} \cos t \\ \sin t \end{array}) 脳 (\begin{matrix} 0 & 4 \\ 5 & 鈭� 3 \end{matrix}) (\begin{array}{l} 鈭� \sin t \\ \cos t \end{array}) \\ 鈭� f (t) = 15 (2 \sin^{2} t 鈭� 1) \end{array}$

The graph of the function is,

Compute the number

Consider the function.

$f (t) = 15 (2 \sin^{2} t 鈭� 1)$

When,. $f (t) = 0 鈬� 15 (2 \sin^{2} t 鈭� 1) = 0$

$t = \frac{蟺}{4}$

Therefore, the number is, . $c = \frac{蟺}{4}$

Compute the perpendicular vectors

Consider the vectors.

$\begin{array}{l} \overset{鈫�}{v_{1}} = (\begin{array}{l} \cos \frac{蟺}{4} \\ \sin \frac{蟺}{4} \end{array}) = \frac{1}{\sqrt{2}} (\begin{array}{l} 1 \\ 1 \end{array}) \\ \overset{鈫�}{v_{2}} = (\begin{array}{l} 鈭� \sin \frac{蟺}{4} \\ \cos \frac{蟺}{4} \end{array}) = \frac{1}{\sqrt{2}} (\begin{array}{l} 鈭� 1 \\ 1 \end{array}) \end{array}$

Consider the linear transformations.

$\begin{array}{l} T (\overset{鈫�}{v_{1}}) = (\begin{matrix} 0 & 4 \\ 5 & 鈭� 3 \end{matrix}) \frac{1}{\sqrt{2}} (\begin{array}{l} 1 \\ 1 \end{array}) = \frac{1}{\sqrt{2}} (\begin{array}{l} 4 \\ 2 \end{array}) \\ T (\overset{鈫�}{v_{2}}) = (\begin{matrix} 0 & 4 \\ 5 & 鈭� 3 \end{matrix}) \frac{1}{\sqrt{2}} (\begin{array}{l} 鈭� 1 \\ 1 \end{array}) = \frac{1}{\sqrt{2}} (\begin{array}{l} 4 \\ 鈭� 8 \end{array}) \end{array}$

Verify the condition.

$\begin{matrix} T (\overset{鈫�}{v_{1}}) 脳 T (\overset{鈫�}{v_{2}}) = \frac{1}{\sqrt{2}} (\begin{array}{l} 4 \\ 2 \end{array}) 脳 \frac{1}{\sqrt{2}} (\begin{array}{l} 4 \\ 鈭� 8 \end{array}) \\ = \frac{1}{2} (\begin{array}{l} 4 \\ 2 \end{array}) 脳 (\begin{array}{l} 4 \\ 鈭� 8 \end{array}) \\ = 0 \end{matrix}$

Therefore, the vectors and their linear transformations are perpendicular.

The graph of the same is,

Hence, the perpendicular unit vectors are,. $\overset{鈫�}{v_{1}} = \frac{1}{\sqrt{2}} (\begin{array}{l} 1 \\ 1 \end{array}), \overset{鈫�}{v_{2}} = \frac{1}{\sqrt{2}} (\begin{array}{l} 鈭� 1 \\ 1 \end{array})$

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91影视

Short Answer

Step by step solution

Compute the function.

Compute the number

Compute the perpendicular vectors

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