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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q47E (page 1)

Consider a linear transformation T from $R^{2} to R^{2}$ . We are told that the matrix T with respect to the basis $[\begin{matrix} 0 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 0 \end{matrix}] is (\begin{matrix} a & b \\ c & d \end{matrix})$
Find the standard matrix of T in terms of a, b, c and d.

Short Answer

Expert verified

The standard matrix of T in terms of a, b, c and d is $A = (\begin{matrix} d & c \\ b & a \end{matrix})$ .

Step by step solution

01

Write the given linear transformation with respect to given basis.

Let $B = ({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2})$ be the basis of $R^{2}$ with

${\overset{鈫�}{v}}_{1} = [\begin{matrix} 0 \\ 1 \end{matrix}] and {\overset{鈫�}{v}}_{2} = [\begin{matrix} 1 \\ 0 \end{matrix}] 鈬� B = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) & B^{- 1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$

Furthermore, let $T : R^{2} 鈫� R^{2}$ be a linear transformation.

The matrix representation of T with respect to B is,

${[T]}_{B} = (\begin{matrix} a & b \\ c & d \end{matrix}) = B (say)$

02

Finding the standard matrix of T

Let A be the standard matrix of T. Then,

$AB = B B 鈬� A = B {BB}^{- 1} 鈬� A = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) 鈬� A = (\begin{matrix} d & c \\ b & a \end{matrix})$

03

Final answer

The standard matrix of T in terms of a, b, c and d is:

$A = (\begin{matrix} d & c \\ b & a \end{matrix})$

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Most popular questions from this chapter

In Exercises 1 through 12, find all solutions of the equations

with paper and pencil using Gauss鈥揓ordan elimination.

Show all your work.

$| \begin{matrix} 2 x_{1} - 3 x_{3} + 7 x_{5} + 7 x_{6} = 0 \\ - 2 x_{1} + x_{2} + 6 x_{5} - 12 x_{6} = 0 \\ x_{2} - 3 x_{3} + x_{5} + 5 x_{6} = 0 \\ - 2 x_{2} + x_{4} + x_{5} + x_{6} = 0 \\ 2 x_{1} + x_{2} - 3 x_{3} + 8 x_{5} + 7 x_{6} = 0 \end{matrix} |$

Three merchants find a purse lying in the road. One merchant says, 鈥淚f I keep the purse, I will have twice as much money as the two of you together.鈥� 鈥淕ive me the purse and I will have three times as much as the two of you together,鈥� said the second merchant. The third merchant said, 鈥淚 will be much better off than either of you if I keep the purse, I will have five times as much as the two of you together.鈥� If there are coins (of equal value) in the purse, how much money does each merchant have? (From Mahavira)

Let be an orthogonal 2X2 matrix. Use the image of the unit circle to find the singular values of A.

Recall that a real square matrix A is called skew symmetric if $A^{T = - A}$ .

a. If A is skew symmetric, is $A^{2}$ skew symmetric as well? Or is $A^{2}$ symmetric?

b. If is skew symmetric, what can you say about the definiteness of $A^{2}$ ? What about the eigenvalues of $A^{2}$ ?

c. What can you say about the complex eigenvalues of a skew-symmetric matrix? Which skew-symmetric matrices are diagonalizable over $鈩�$ ?

In Exercises 1 through 12, find all solutions of the equations

with paper and pencil using Gauss鈥揓ordan elimination.

Show all your work.

$| \begin{matrix} x_{4} + 2 x_{5} - x_{6} = 2 \\ x_{1} + 2 x_{2} + x_{5} - x_{6} = 0 \\ x_{1} + 2 x_{2} + 2 x_{3 -} x_{5} - x_{6} = 2 \end{matrix} |$

See all solutions

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