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.

18. $T (f) = 2 f + 3 f' from P_{2} to P_{2}$

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

$d e t T = d e t B = 8$ .

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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,

$T (f) = 2 f + 3 f' from P_{2} to P_{2}$

Step 3: To find determinant.

Forwe $f \in P_{2}$ have $f (t) = a t^{2} + b t + c$ . So we compute

$T (f) (t) = 2 f (t) + 3 f' (t) T (f) (t) = 2 (a t^{2} + b t + c) + 3 (2 a t + b) T (f) (t) = 2 a t^{2} + (6 a + 2 b) t + (3 b + 2 c)$

Since $B = \{t^{2}, t, 1\}$ is a basis for $P_{2}$ , the matrix of T corresponding to B is

$B = [\begin{matrix} 2 & 0 & 0 \\ 6 & 2 & 0 \\ 0 & 3 & 2 \end{matrix}] \begin{matrix} \end{matrix}$

Therefore,

det T =det B = 8.

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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.

18. $T (f) = 2 f + 3 f' from P_{2} to P_{2}$

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. 18. T(f)=2f+3f' from P2 to P2

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.

18. $T (f) = 2 f + 3 f' from P_{2} to P_{2}$