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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 6: Q17E (page 289)

Find the determinants of the linear transformations in Exercises 17 through 28.
17. $T (f) = 2 f + 3 f' from P_{2} to P_{2}$

Short Answer

Expert verified

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

det T = det B = 8 .

Step by step solution

01

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.

02

Given

Given linear transformation,

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

03

To find determinant

For $f 鈭� P_{2}$ we 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

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

Therefore,

det T =det B = 8.

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Most popular questions from this chapter

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

4. $[\begin{matrix} 1 & - 1 & 2 & - 2 \\ - 1 & 2 & 1 & 6 \\ 2 & 1 & 14 & 10 \\ - 2 & 6 & 10 & 33 \end{matrix}]$

For every nonzero $2 脳 2$ matrix A there exists a $2 脳 2$ matrix B such that $d e t (A + B) 鈮� d e t A + d e t B$ .

Use Cramer's rule to solve the systems in Exercises 22 through 24.

23. $| \begin{matrix} 5 x_{1} & - 3 x_{2} & = 1 \\ - 6 x_{1} & + 7 x_{2} & = 0 \end{matrix} |$

a. Find a noninvertible $2 脳 2$ matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible $3 脳 3$ matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

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

22. $T (f (t)) = f (- t) f r o m P_{n} t o P_{n}$

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