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.

21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$

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

det T = det B = 1

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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,

$T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$

Step 3: To find determinant.

Consider the linear transformation $T : P_{3} \to P_{3}$ defined as $T (f) = f (- t)$ .

Consider the basis $B = \{1, t, t^{2}, t^{3}\}$ for $P_{3}$ .

We have that

$T (1) = 1 T (1) = 1 (1) + 0 t + 0 t^{2} + 0 t^{3} T (t) = - t T (t) = 0 (1) + (- 1) t + 0 t^{2} + 0 t^{3} T (t^{2}) = {(- t)}^{2} T (t^{2}) = t^{2} = 0 (1) + 0 t + 1 t^{2} + 0 t^{3} a n d T (t^{3}) = {(- t)}^{3} T (t^{3}) = - t^{3} T (t^{3}) = 0 (1) + 0 t + 0 t^{2} + (- 1) t^{3}$

Thus the matrix of T with respect tois

$B = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}]$

Observe that

$d e t B = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}] d e t B = 1$

Therefore,

det T = det B =1

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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.

21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$

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. 21. T(f(t))=f(-t) from P3 to P3

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.

21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$