Let P2 denote the vector space of polynomials of degree up to 2. Which of the following subsets of P2 are subspaces of P2? ]Ā. {p(t) | p'(1) = p(2)} B. {p(t) | ³p(t)dt = 0} C. {p(t) |p(5) = 7} D. {p(t) | p'(t) is constant } E. {p(t) | p(-t) = -p(t) for all t} OF. {p(t) | p' (t) + 4p(t) + 2 = 0}
Let P2 denote the vector space of polynomials of degree up to 2. Which of the following subsets of P2 are subspaces of P2? ]Ā. {p(t) | p'(1) = p(2)} B. {p(t) | ³p(t)dt = 0} C. {p(t) |p(5) = 7} D. {p(t) | p'(t) is constant } E. {p(t) | p(-t) = -p(t) for all t} OF. {p(t) | p' (t) + 4p(t) + 2 = 0}
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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![Let P₂ denote the vector space of polynomials of degree up to 2.
Which of the following subsets of P₂ are subspaces of P₂?
]Ā. {p(t) | p′(1) = p(2)}
3
B. {p(t) | √³ p(t)dt = 0}
C. {p(t) | p(5) = 7}
D. {p(t) | p'(t) is constant}
Ē. {p(t) | p(−t) = -p(t) for all t}
Ƒ. {p(t) | p'(t) + 4p(t) + 2 = 0}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd29ed1ca-eabb-4fe1-ba2f-df7a36f73903%2Fbdf5b42a-feb2-4428-9b0d-abf25c680fbe%2Fqi4z4ib_processed.png&w=3840&q=75)
Transcribed Image Text:Let P₂ denote the vector space of polynomials of degree up to 2.
Which of the following subsets of P₂ are subspaces of P₂?
]Ā. {p(t) | p′(1) = p(2)}
3
B. {p(t) | √³ p(t)dt = 0}
C. {p(t) | p(5) = 7}
D. {p(t) | p'(t) is constant}
Ē. {p(t) | p(−t) = -p(t) for all t}
Ƒ. {p(t) | p'(t) + 4p(t) + 2 = 0}
![Determine whether the given set S is a subspace of the vector space V.
A. V = P₂, and S is the subset of P2₂ consisting of all polynomials of the form p(x) = x² + c.
B. V = P3, and S is the subset of P3 consisting of all polynomials of the form p(x) = ax³ + bx.
OC. V is the vector space of all real-valued functions defined on the interval [a, b], and S is the subset of V consisting of those functions satisfying f(a) = 7.
OD. V is the space of three-times differentiable functions R → R, and S is the subset of V consisting of those functions satisfying the differential equation y"" + 6y = x².
E. V = RXn, and S is the subset of all skew-symmetric matrices.
OF. V = R², and S is the set of all vectors (x₁, x2) in V satisfying 7x₁ + 8x2
G. V is the vector space of all real-valued functions defined on the interval (-∞, ∞), and S is the subset of V consisting of those functions satisfying f(0) = 0.
=
0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd29ed1ca-eabb-4fe1-ba2f-df7a36f73903%2Fbdf5b42a-feb2-4428-9b0d-abf25c680fbe%2F2ud37s8_processed.png&w=3840&q=75)
Transcribed Image Text:Determine whether the given set S is a subspace of the vector space V.
A. V = P₂, and S is the subset of P2₂ consisting of all polynomials of the form p(x) = x² + c.
B. V = P3, and S is the subset of P3 consisting of all polynomials of the form p(x) = ax³ + bx.
OC. V is the vector space of all real-valued functions defined on the interval [a, b], and S is the subset of V consisting of those functions satisfying f(a) = 7.
OD. V is the space of three-times differentiable functions R → R, and S is the subset of V consisting of those functions satisfying the differential equation y"" + 6y = x².
E. V = RXn, and S is the subset of all skew-symmetric matrices.
OF. V = R², and S is the set of all vectors (x₁, x2) in V satisfying 7x₁ + 8x2
G. V is the vector space of all real-valued functions defined on the interval (-∞, ∞), and S is the subset of V consisting of those functions satisfying f(0) = 0.
=
0.
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