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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: Q19E (page 289)

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

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.

$d e t T = d e t B = - 1$ Consider the linear transformation $T : P_{2} 鈫� P_{2}$ defined as $T (f) = f (- t)$ .

Consider the basisfor $B = \{1, t, t^{2}\} for P_{2}$ .
We have that

$T (1) = 1 T (1) = 1 (1) + 0 t + o t^{2}, T (t) = - t T (t) = 0 (1) + (- 1) t + 0 t^{2}, T (t^{2}) = {(- t)}^{2} + t^{2} T (t^{2}) = 0 (1) + 0 t + 1 t^{2} .$

Thus the matrix of T with respect to B is

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

Observe that

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

Therefore,

det T =det B = -1

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Most popular questions from this chapter

Is the determinant of the matrix

$A = [\begin{matrix} 1 & 1000 & 2 & 3 & 4 \\ 5 & 6 & 7 & 1000 & 8 \\ 1000 & 9 & 8 & 7 & 6 \\ 5 & 4 & 3 & 2 & 1000 \\ 1 & 2 & 1000 & 3 & 4 \end{matrix}]$

positive or negative? How can you tell? Do not use technology.

Consider the function $F (A) = F [\overset{鈫�}{v} \overset{鈫�}{w}] = \overset{鈫�}{v} 脳 \overset{鈫�}{w}$ from ${鈩�}^{2 脳 2}$ to $鈩�$ , the dot product of the column vectors of A.
a. Is Flinear in both columns of A? See Example 6.
b. Is F linear in both rows of A?
c. Is Falternating on the columns of A? See Example 4.

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

9. $[\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 1 & 3 & 3 & 3 \\ 1 & 1 & 1 & 4 & 4 \\ 1 & 1 & 1 & 1 & 5 \end{matrix}]$

For an invertiblenxnmatrix A, what is adj(adj A)?

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

See all solutions

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