Let S = {p(x) = a + bx|a+b=0}. a) Show that S is a subspace of P2. [Hint: use theorem 3 or 4 of section 3.2]. Explain clearly. b) Find a basis for S. Specify the dimension of S. [Hint: solve a + b = 0 for a; then substitute into p(x) and collect like terms]
Let S = {p(x) = a + bx|a+b=0}. a) Show that S is a subspace of P2. [Hint: use theorem 3 or 4 of section 3.2]. Explain clearly. b) Find a basis for S. Specify the dimension of S. [Hint: solve a + b = 0 for a; then substitute into p(x) and collect like terms]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Help with question 5
![5. Let S = {p(x) = a + bx|a+b=0}.
a) Show that S is a subspace of P2. [Hint: use theorem 3 or 4 of section 3.2]. Explain
clearly.
b) Find a basis for S. Specify the dimension of S. [Hint: solve a + b = 0 for a; then
substitute into p(x) and collect like terms]
6. If
and
17. Let
explain why span(S) = span(T).
and
bas e
and
S =
T=
5
10. Let B =
1
2
s={[4] [B]}
S
"
5
(Hint: see Example 9 pg. 170)
9. Let
2
3
= {[4].[1].[B]}
5
2
1
0
{····}
-4
2
6
0
0
5
Find a basis for the span of S. (Hint: see Example 8 pg. 169)
S
8. Find a basis for the vector space V = R³ that contains the vectors in
5
0
1(3)
{[B][B]}
10
=
3
1
- {[@]-[-2]}
{
Igialura talaba bus noftibly, b
10
B =
1
B' =
1
{@[27]}
1
1
vector [v]B for v relative to basis B.
-2
3
-1
4
- {[2¹] [3]}
>
be two ordered bases for R2. Find the change of basis matrix [I]B.
sqedire's (5)
5
10
be a basis for R2, and let v = [18]
15
ob
(d
Find the coordinate
** Bonus Question: (+5 points) Let S and T be the subspaces of P3 be defined by
S = {p(x) |p(0) = 0}
U
T = {q(x) | q(1) = 0}
Find dim(SNT). [Hint: characterize polynomials in the intersection of S and T]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8783cbc5-0f10-43ba-8bd7-4625ee264f91%2F7a38895a-5e69-4760-a04f-7e5f9e9cd847%2Fhuw9ffk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. Let S = {p(x) = a + bx|a+b=0}.
a) Show that S is a subspace of P2. [Hint: use theorem 3 or 4 of section 3.2]. Explain
clearly.
b) Find a basis for S. Specify the dimension of S. [Hint: solve a + b = 0 for a; then
substitute into p(x) and collect like terms]
6. If
and
17. Let
explain why span(S) = span(T).
and
bas e
and
S =
T=
5
10. Let B =
1
2
s={[4] [B]}
S
"
5
(Hint: see Example 9 pg. 170)
9. Let
2
3
= {[4].[1].[B]}
5
2
1
0
{····}
-4
2
6
0
0
5
Find a basis for the span of S. (Hint: see Example 8 pg. 169)
S
8. Find a basis for the vector space V = R³ that contains the vectors in
5
0
1(3)
{[B][B]}
10
=
3
1
- {[@]-[-2]}
{
Igialura talaba bus noftibly, b
10
B =
1
B' =
1
{@[27]}
1
1
vector [v]B for v relative to basis B.
-2
3
-1
4
- {[2¹] [3]}
>
be two ordered bases for R2. Find the change of basis matrix [I]B.
sqedire's (5)
5
10
be a basis for R2, and let v = [18]
15
ob
(d
Find the coordinate
** Bonus Question: (+5 points) Let S and T be the subspaces of P3 be defined by
S = {p(x) |p(0) = 0}
U
T = {q(x) | q(1) = 0}
Find dim(SNT). [Hint: characterize polynomials in the intersection of S and T]
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