Q21E Find the determinants of the lin... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q21E (page 289)

Find the determinants of the linear transformations in Exercises 17 through 28.
21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$

Short Answer

Expert verified

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

det T = det B = 1

Step by step solution

Definition.

A determinant is a unique number associatedwith 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.

Given.

Given linear transformation,

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

To find determinant.

Consider the linear transformation $T : P_{3} 鈫� P_{3}$ 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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91影视

Find the determinants of the linear transformations in Exercises 17 through 28.
21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$

Short Answer

Step by step solution

Definition.

Given.

To find determinant.

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91影视

Find the determinants of the linear transformations in Exercises 17 through 28.21.T(f(t))=f(-t)fromP3toP3

Short Answer

Step by step solution

Definition.

Given.

To find determinant.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Calculus

Geometry

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the determinants of the linear transformations in Exercises 17 through 28.
21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$