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Assignment 5

1. Use Theorem 4.2 to determine which of the following are subspaces.

(a) The set of all polynomials a0 + a1x + a2x2 + a3x3 for which a0 + a1 + a2 + a3 = 0.

(b) The set of all polynomials a0 + a1x + a2x2 + a3x3 for which a0, a1, a2, and a3 are integers.

(c) The set of all polynomials a0 + a1x + a2x2 + a3x3 for which a1 × a3 = 0.

(d) The set of all vectors in R3 with the first coordinate component nonzero.

(e) The set of all diagonal matrices in Rn×n.

(f) The set of all vectors x in R n such that Ax = b, where A ∈ R n×n and b ≠ 0.

(g) The set of all differentiable functions f = f(x) in F(−∞, +∞) that satisfy

2. Determine whether the vector v = [0, 5, 6, −3] is contained in the subspace spanned by u1, u2, u3, and u4, where

u1 = [−1, 3, 2, 0],   u2 = [2, 0, 4, −1],   u3 = [7, 1, 1, 4],   u4 = [6, 3, 1, 2].

3. Let A(1), A(2), A(3), and B be matrices in R2×2, where

Determine whether B is contained in span{A(1), A(2), A(3)}.

4. Let W be the set of all vectors of each given form, where a, b, and c represent arbitrary real numbers. Determine whether W is a subspace of R4 . If so, find a set S of vectors that spans W.

(a) [2a + 3b, −1, 2a − 5b, 5a].   (b) [2a − b, 3b − c, 3c − a, 3b].

5. Let u, v, and w be linearly independent vectors in a vector space V . Show that

(a) {u, u + v, u + v + w} is linearly independent.

(b) {u + v, u + w, v + w} is linearly independent.

(c) {u − v, u − w, v − w} is linearly dependent.