# Math homework

Math 2568 Autumn 2020

Homework 3

In the first six problems determine whether or not the set given is a subspace. In each case give a justification for your answer.

Problem 1 Let W = {x ∈ R3} satisfying the equation x21 = x2 + x3.

Problem 2 Let W = {x ∈ R3} satisfying the equations x1 = 3×3, x2 = −4×3.

Problem 3 Let W = {x ∈ R3} satisfying the equation x1x2 = x3.

Problem 4 Let a, b be fixed vectors in R3, and W = {x ∈ R3} satisfying the equation xT ∗ a = xT ∗ b.

Problem 5 Let W = {x ∈ R3} satisfying the inequality x1 ≥ x2.

Problem 6 Let A ∈ R3×3 be a fixed 3 × 3 matrix, and W = {v ∈ R3} satisfying the property that the equation A ∗ x = v is consistent.

In the following four problems determine it the statement is true of false. In either case give a justification for your answer.

Problem 7 Let U, V be non-zero subspaces of R3, and let W = U ∪ V be the set of vectors which lie in either U or V (or both). Then W is a subspace of R3.

Problem 8 Let U, V be non-zero subspaces of R3, and let W = U ∩ V be the set of vectors which lie in both U and V . Then W is a subspace of R3.

Problem 9 Let U, V be non-zero subspaces of R3, and let W = U + V be the set of vectors w ∈ R3 which can be written as a sum w = u + v with u ∈ U,v ∈ V . Then W is a subspace of R3.

Problem 10 Let W = {x ∈ R3} satisfying x21 + x22 + x23 ≥ 0. Then W is a subspace of R3.

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