P(-2) p(3) p(1) p(0) 10. Define T : P3 → R* by T (p) = a. Show that T is a linear transformation. b. Find the matrix for T relative to the basis {1,t,t²,t³} for P3 and the standard basis for R*.
P(-2) p(3) p(1) p(0) 10. Define T : P3 → R* by T (p) = a. Show that T is a linear transformation. b. Find the matrix for T relative to the basis {1,t,t²,t³} for P3 and the standard basis for R*.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Question
Number 10 from 5.4
![22. If A is diagonalizable anc
diagonalizable.
9. Define T : P2 → R³ by T(p) =
p(0)
P(1)
a. Find the image under T of p(t) = 5+ 3t.
23. If B = P-'AP and x is a
to an eigenvalue 1, then F
sponding also to 2.
b. Show that T is a linear transformation.
c. Find the matrix for T relative to the basis {1,t,t2} for P2
and the standard basis for R3.
24. If A andB are similar, the
Refer to Supplementary E-
P(-2)
p(3)
p(1)
p(0)
a. Show that T is a linear transformation.
25. The trace of a square ma
10. Define T : P3 → R* by T(p) =
entries in A and is denote
tr(FG) = tr(GF) for an-
Show that if A and B are s
b. Find the matrix for T relative to the basis {1, t,t²,t³} for
P3 and the standard basis for R*.
26. It can be shown that the tra
the eigenvalues of A. Verif
A is diagonalizable.
In Exercises 11 and 12, find the B-matrix for the transformation
XH Ax, where B = {b¡,b2}.
27. Let V be R" with a basis
with the standard basis, de
identity transformation I
the matrix for I relative t
-4
11. A =
6.
-1
b2 =
called in Section 4.4?
-6
12. A =
4
b, =
b, =
28. Let V be a vector space w
be the same space V with
In Exercises 13–16, define T : R² → R² by T(x) = Ax. Find a
basis B for R? with the property that [T]B is diagonal.
be the identity transforma
for I relative to B and C
Section 4.7?
13. А 3
14. A =
-3
29. Let V be a vector space E
the B-matrix for the iden
-2
15. A =
16. A =
[M] In Exercises 30 and 31, fi
tion x > Ax where B = {b1,
17. Let A =
and B = {b1, b2}, for b¡ =
6 -2 -2
1 -2
30. A =
b2 =
Define T: R2 → R² by T(x) = Ax.
2 -2
2
17
, b2 =| 1
[27
a. Verify that bị is an eigenvector of A but that A is not
diagonalizable.
bj =
b. Find the B-matrix for T.
1
3
-7 -48 -16
6.
18. Define T: R³ → R³ by T (x)
matrix with eigenvalues 5, 5, and -2. Does there exist a basis
B for R3 such that the B-matrix for T is a diagonal matrix?
Discuss.
= Ax, where A is a 3 x 3
31. A=
1
14
-3 -45 -19
-3
b =
1
,b2 D
Verify the statements in Exercises 19-24. The matrices are square.
-3
19. If A is invertible and similar to B, then B is invertible
and A- is similar to B-. [Hint: P-AP = B for some
invertible P. Explain why B is invertible. Then find an
invertible Q such that 0-A-0 = R-1
32. [M] Let T be the transf
given below. Find a basis
is diagonal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2F9fcc3451-13ec-4d00-a7a4-bfd3fc2df301%2Fivqri6c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:22. If A is diagonalizable anc
diagonalizable.
9. Define T : P2 → R³ by T(p) =
p(0)
P(1)
a. Find the image under T of p(t) = 5+ 3t.
23. If B = P-'AP and x is a
to an eigenvalue 1, then F
sponding also to 2.
b. Show that T is a linear transformation.
c. Find the matrix for T relative to the basis {1,t,t2} for P2
and the standard basis for R3.
24. If A andB are similar, the
Refer to Supplementary E-
P(-2)
p(3)
p(1)
p(0)
a. Show that T is a linear transformation.
25. The trace of a square ma
10. Define T : P3 → R* by T(p) =
entries in A and is denote
tr(FG) = tr(GF) for an-
Show that if A and B are s
b. Find the matrix for T relative to the basis {1, t,t²,t³} for
P3 and the standard basis for R*.
26. It can be shown that the tra
the eigenvalues of A. Verif
A is diagonalizable.
In Exercises 11 and 12, find the B-matrix for the transformation
XH Ax, where B = {b¡,b2}.
27. Let V be R" with a basis
with the standard basis, de
identity transformation I
the matrix for I relative t
-4
11. A =
6.
-1
b2 =
called in Section 4.4?
-6
12. A =
4
b, =
b, =
28. Let V be a vector space w
be the same space V with
In Exercises 13–16, define T : R² → R² by T(x) = Ax. Find a
basis B for R? with the property that [T]B is diagonal.
be the identity transforma
for I relative to B and C
Section 4.7?
13. А 3
14. A =
-3
29. Let V be a vector space E
the B-matrix for the iden
-2
15. A =
16. A =
[M] In Exercises 30 and 31, fi
tion x > Ax where B = {b1,
17. Let A =
and B = {b1, b2}, for b¡ =
6 -2 -2
1 -2
30. A =
b2 =
Define T: R2 → R² by T(x) = Ax.
2 -2
2
17
, b2 =| 1
[27
a. Verify that bị is an eigenvector of A but that A is not
diagonalizable.
bj =
b. Find the B-matrix for T.
1
3
-7 -48 -16
6.
18. Define T: R³ → R³ by T (x)
matrix with eigenvalues 5, 5, and -2. Does there exist a basis
B for R3 such that the B-matrix for T is a diagonal matrix?
Discuss.
= Ax, where A is a 3 x 3
31. A=
1
14
-3 -45 -19
-3
b =
1
,b2 D
Verify the statements in Exercises 19-24. The matrices are square.
-3
19. If A is invertible and similar to B, then B is invertible
and A- is similar to B-. [Hint: P-AP = B for some
invertible P. Explain why B is invertible. Then find an
invertible Q such that 0-A-0 = R-1
32. [M] Let T be the transf
given below. Find a basis
is diagonal.
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