Q63E Consider the linear transformati... [FREE SOLUTION]

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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 7: Q63E (page 359)

Consider the linear transformation T ( f ) = f from C鈭� to $\frac{- b 卤 \sqrt{b^{2} - 4 ac}}{2 a}$ C鈭�. For each of the following eigenvalues, find a basis of the associated eigenspace. See Exercise 62. a. 位 = 1 b. 位 = 0 c. 位 = 鈭�1 d. 位 = 鈭�4.

Short Answer

Expert verified

$a . E_{1} = span (\{e^{x}, e^{- x}\}) b . E_{0} = span (\{x, 1\}) c . E_{- 1} = span (\{\cos, \sin\}) d . E_{- 4} = span (\{\sin (2 x), \cos (2 x)\})$

Step by step solution

(a) Step 1: For λ=1

We solve,

$\begin{array}{l} f^{''} (x) = {x) \\ f (x) = C_{2} - e^{x} + C_{1}, e 脳 \\ E_{1} = s p a n (\{e^{x}, e^{- x}\}) \end{array}$

(b) Step 2:For λ=0

We solve,

$\begin{array}{l} f^{''} (x) = {x) \\ f (x) = C_{2} x + C_{1} \\ E_{0} = span (\{x, 1\}) \end{array}$

(c) Step 3:For λ=-1

We solve,

$\begin{array}{l} f^{''} (x) = - f {x) \\ f (x) = C_{2} sinx + C_{1} cosx \\ E_{- 1} = span (\{\cos, \sin\}) \end{array}$

(d) Step 4: For λ=-4λ=-4

We solve,

$\begin{array}{l} f^{''} (x) = - 4 f {x) \\ f (x) = C_{2} \cos (2 x) + C_{1} sinx (2 x) \\ E_{- 4} = span (\{\sin (2 x), \cos (2 x)\}) \end{array}$

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91影视

Consider the linear transformation T ( f ) = f from C鈭� to $\frac{- b 卤 \sqrt{b^{2} - 4 ac}}{2 a}$ C鈭�. For each of the following eigenvalues, find a basis of the associated eigenspace. See Exercise 62. a. 位 = 1 b. 位 = 0 c. 位 = 鈭�1 d. 位 = 鈭�4.

Short Answer

Step by step solution

(a) Step 1: For λ=1

(b) Step 2:For λ=0

(c) Step 3:For λ=-1

(d) Step 4: For λ=-4λ=-4

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91影视

Consider the linear transformation T ( f ) = f from C鈭� to-b卤b2-4ac2aC鈭�. For each of the following eigenvalues, find a basis of the associated eigenspace. See Exercise 62. a. 位 = 1 b. 位 = 0 c. 位 = 鈭�1 d. 位 = 鈭�4.

Short Answer

Step by step solution

(a) Step 1: For λ=1

(b) Step 2:For λ=0

(c) Step 3:For λ=-1

(d) Step 4: For λ=-4λ=-4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Logic and Functions

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Consider the linear transformation T ( f ) = f from C鈭� to $\frac{- b 卤 \sqrt{b^{2} - 4 ac}}{2 a}$ C鈭�. For each of the following eigenvalues, find a basis of the associated eigenspace. See Exercise 62. a. 位 = 1 b. 位 = 0 c. 位 = 鈭�1 d. 位 = 鈭�4.