Q50E In Exercises 48 through 53, let ... [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 4: Q50E (page 196)

In Exercises 48 through 53, let V be the space spanned by the two functions $c o s (t)$ and $s i n (t)$ . In each exercise, find the matrix of the given transformation T with respect to the basis $c o s (t), s i n (t)$ , and determine whether T is an isomorphism.
50. $T (f) = f'' + a f' + b f$ ,where a and b are arbitrary real numbers.Find all the values of a and b such that T is an isomorphism

Short Answer

Expert verified

The solution is $B = (\begin{matrix} b 鈭� 1 & a \\ 鈭� a & b 鈭� 1 \end{matrix})$

Step by step solution

Step 1:Solution for the matrix of the given transformation T

Consider the matrix of the given transformation be T

Let $T (f) = f " + a f' + b f$ be the transformation.

Assume $f (x) = a \cos (x) + b \sin (x)$ be the function.

Hence $B = [{[T (\cos x)]}_{B} 鈥夆赌夆赌� {[T (\sin x)]}_{B} 鈥�]$

Then, the transformation be as follows.

$\begin{matrix} T \cos (x) = 鈭� \cos (x) 鈭� a \sin (x) + b \cos (x) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆€� = (b 鈭� 1) \cos (x) 鈭� a \sin (x) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆€� \end{matrix}$

Similarly further simplification is as follows

$\begin{matrix} T (\sin x) = 鈭� \sin (x) + a \cos (x) + b \sin (x) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€� = (b 鈭� 1) \sin (x) + a \cos (x) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€� = a \cos (x) + (b 鈭� 1) \sin (x) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€� \end{matrix}$

Therefore the matrix be B as follows

$B = (\begin{matrix} b 鈭� 1 & a \\ 鈭� a & b 鈭� 1 \end{matrix})$

Hence the solution.

Step 2:Solution for the isomorphism of the given transformation T

Consider $B = (\begin{matrix} b 鈭� 1 & a \\ 鈭� a & b 鈭� 1 \end{matrix})$ be the matrix of the given transformation T.

$\begin{matrix} [B] = \det (B) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆€� = a d 鈭� b c \\ 鈥夆赌夆赌夆€夆赌夆赌夆€夆€� = (b 鈭� 1) (b 鈭� 1) 鈭� (a) (鈭� a) \\ 鈥夆赌夆赌夆€夆赌夆赌夆€� = b^{2} 鈭� 2 b + (a^{2} 鈭� 1) 鈮� 0 \end{matrix}$

Thus the transformation T is invertible as well.So T is an isomorphism.

T is isomorphism on all values other than $a = 1, - 1 a n d b = 2$ .

Hence the solution.

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91影视

Short Answer

Step by step solution

Step 1:Solution for the matrix of the given transformation T

Step 2:Solution for the isomorphism of the given transformation T

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