Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the determinants of the linear transformations in Exercises 17 through 28.

27. $T (f) = a f' + b f "$ , where a and b are arbitrary constants, from the space V spanned by $c o s (x)$ and $s i n (x)$ to V

Therefore, the determinant of the linear transformations is given by,

$d e t T = d e t B = a^{2} + b^{2}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition.

A determinant is a unique number associated with a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Step 2: Given.

Given linear transformation,

$T (f) = a f' + b f "$

Step 3: To find determinant.

For $f \in V$ , we have $f (x) = α \cos (x) + β \sin (x)$ .

So we compute

T (f) (x) = a f' (x) + b f " (x) T (f) (x) = a (- α \sin (x) + β \cos (x)) + b (- α \cos (x) - β \sin (x)) T (f) (x) = (- b α + a β) \cos (x) + (- a α - b β) \sin (x)

Obviously $B = \{\cos (x), \sin (x)\}$ is a basis for $V$ , so the matrix of $T$ that corresponds to $B$ is

$B = [\begin{matrix} - b & a \\ - a & - b \end{matrix}]$

Therefore,

$d e t T = d e t B = a^{2} + b^{2}$ .

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Linear Algebra With Applications

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Short Answer

Find the determinants of the linear transformations in Exercises 17 through 28.

27. $T (f) = a f' + b f "$ , where a and b are arbitrary constants, from the space V spanned by $c o s (x)$ and $s i n (x)$ to V

Step by Step Solution

Step 1: Definition.

Step 2: Given.

Step 3: To find determinant.

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Linear Algebra With Applications

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Short Answer

Find the determinants of the linear transformations in Exercises 17 through 28. 27. T(f)=af'+bf", where a and b are arbitrary constants, from the space V spanned by cos(x) and sin(x)to V

Step by Step Solution

Step 1: Definition.

Step 2: Given.

Step 3: To find determinant.

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Find the determinants of the linear transformations in Exercises 17 through 28.

27. $T (f) = a f' + b f "$ , where a and b are arbitrary constants, from the space V spanned by $c o s (x)$ and $s i n (x)$ to V