3. A) Let W be a subset of $P_2$ given by $W = \{p(x) = ax^2 + bx + c|a, b, c \in \mathbb{R}, p(2) = 1\}$ Is W a subspace of $P_2$? Explain. B) Let V be a subset of $P_2$ given by $V = \{p(x) = ax^2 + bx + c|a, b, c \in \mathbb{R}, p(2) = 0\}$ Is W a subspace of $P_2$? Explain.
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**Closure under addition:** If p(x) and q(x) are in W, then p(x) + q(x) must also be in W. 2. **Closure under scalar multiplication:** If p(x) is in W and k is a scalar, then kp(x) must also be in W. 3. **Contains the zero vector:** The zero vector of P2, which is Show more…
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Let P2 be the vector space of all polynomials of degree 2 or less. We can write this as P2 = {a + bx + cx^2 | a, b, c ∈ R}. Let W be the subset W = {a + a^2 + a^3 | a ∈ R}. a) Write down polynomial p(z) that is in W. b) Write down polynomial q(z) that is in P2 but not in W. c) Is W a subspace of P3? If yes, show that it is closed under addition and scalar multiplication. If not, show that it fails one of these. Repeat all three parts of Exercise #2, but use the subset X of P2 given by X = {a + a^2 + 10a + 10 | a ∈ R}. Repeat all three parts of Exercise #2, but use the subset U of P2 given by U = {a^2 + a^3 + (10 + a)x + a | a ∈ R}.
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a) What is a "subspace" of a vector space V? b) Is W = {f(x): f is a polynomial with degree ≤ 3, f'(0) = 0} a subspace of P₃: all polynomials with degree ≤ 3? Explain your answer. c) Is W = {(x, y): y = 2x²} a subspace of ℝ²? Explain your answer.
Let V be a non-empty vector space. Let W be a non-empty subset of V. What single condition do we use to determine whether W is a subspace of V? Let V = P2(R), the vector space of polynomials of degree less than or equal to 2. (An element of V is of the form p(x) = a0 + a1x + a2x^2.) (a) Is Z = {p(x) ∈ V : p(2) = 0} a subspace of V? (Show your work.) (b) Is U = {p(x) ∈ V : p(2) = 3} a subspace of V? (Show your work.)
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