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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: Q18E (page 289)

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

Short Answer

Expert verified

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

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

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.

Forwe $f 鈭� P_{2}$ 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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Most popular questions from this chapter

$d e t (A + B) = d e t A + d e t B$ for all $5 脳 5$ matricesA andB.

Which of the following functions F of $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ are linear in both columns? Which are linear in both rows? Which are alternating on the columns?
role="math" localid="1659502796300" $a . F (A) = b c b . F (A) = c d c . F (A) = a c d . F (A) = b c - a d e . F (A) = c$

If Ais an invertible $n 脳 n$ matrix, then $d e t (A^{T})$ must equal $d e t (A^{- 1})$ .

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

3. $[\begin{matrix} 1 & 3 & 2 & 4 \\ 1 & 6 & 4 & 8 \\ 1 & 3 & 0 & 0 \\ 2 & 6 & 4 & 12 \end{matrix}]$

There exists a nonzero $4 脳 4$ matrixAsuch that $d e t A = d e t A (4 A)$ .

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