Vector Spaces and Subspaces - Solutions

1. Vector Spaces and Subspaces (References: Comps Study Guide for Linear Algebra Section 1; Damiano & Little, A Course in Linear Algebra, Chapter 1) Vector Spaces Basic Examples of Vector Spaces. For each example, identify: (a) an expression for a general vector in the vector space (b) the definitions of the addition and scalar multiplication in the vector space (c) the zero vector (or the additive identity) in the vector space (d) the additive inverse of a general vector in the vector space 1. Rn 2. Pn (R) 3. Mm⇥n (R) Subspaces: Suppose V is a vector space. Explain what it means to say that a subset U of V is a subspace. Subspace Theorem: Write down the theorem that you use to prove that a subset W of a vector space V is a subspace. Ex. 1. (a) Let V = R2 and W = {(x, y) 2 R2 | 6x + 5y = 3}. Determine whether or not W is a subspace of V and justify your answer. (b) Give two more examples of subsets of R2 : one that is a subspace and one that is not a subspace. Justify your answers. Ex. 2. V = P2 (R) be the vector space of polynomials of degree two or less. Determine whether or not each of the following sets W is a subspace of V . Justify your answers. (a) W = {p 2 P2 (R) | p(0) + p0 (0) = 0} (b) W = {p 2 P2 (R) | p0 (0) = 2} Ex. 3. Let U and V be subspaces of a vector space W . Prove that is a subspace of W . 2U + 5V = {2~u + 5~v : ~u 2 U, ~v 2 V } Linear Combinations: Let V be a vector space and S = {v1 , . . . , vn } be a subset of V . Write down a general linear combination of the elements of S. Span: The span of a nonempty set S is the set of all linear combination of elements of S. Write down the definition of Span(S) in set-builder notation. (Note: Span(;) is defined to be {~0}.) Ex. 4. Let W be a subspace of a vector space V and let S be a subset of W . Prove that Span(S) ✓ W . (Note: Here you are being asked to prove a common theorem, which you could usually use without proof.) Ex. 5. Let V = P2 (R) be the vector space of polynomials with real coefficients of degree at most two and let S = {1, 1 + x, 1 + x + x2 }. Prove that Span(S) = V . Linear Independence and Dependence: Let S = {v1 , . . . , vn } be a set of vectors in a vector space V . The set S is linearly dependent if The set S is linearly independent if Ex. 6. Is the following set of polynomials linearly independent? Explain your answer. {x4 , x4 + x3 , x4 + x3 + x2 } T Ex. 7. Let W1 and W2 be subspaces of a vector space V and assume that W1 W2 = {~0}. Let w1 2 W1 and w2 2 W2 be such that w1 6= ~0 and w2 6= ~0. Prove that {w1 , w2 } is linearly independent. Ex. 8. Suppose that u and v are vectors in a vector space V . Prove that {u, v} is linearly independent if and only if {u + v, u v} is linearly independent. Basis: A subset S of a vector space V is a basis of V if (1) (2) Dimension: The dimension of a vector space V is the number of elements in a basis for V . (It is a theorem that any two bases of V have the same number of elements.) If V has no finite basis, we say dim(V ) = 1. Ex. 9. Let P3 (R) be the vector space of polynomials with real coefficients and degree at most three. Let W = {p 2 P3 (R) | p(0) = p00 (0) and p0 (1) = 0}. (a) Prove that W is a subspace of P3 (R). (b) Find a basis for W . (c) What is the dimension of W ? Ex. 10. (a) Give a basis for the subspace of R4 spanned by the vectors 2 3 2 3 2 3 1 2 0 607 607 617 6 7, 6 7, 6 7, 425 4 45 425 1 2 1 2 3 2 6 17 6 7. 425 1 (b) Give an example of a vector in R4 that is not in the subspace in part (a). Justify your answer. Additional Problems Ex. 11. Determine whether or not each of the following sets W is a subspace of the vector space V . Justify your answers. (a) V = P (R) be the vector space of all polynomials and W = {p 2 P (R) | p(2) = 0}. (b) V = P (R) be the vector space of all polynomials and W = {p 2 P (R) | p(2) = 1}. (c) V = R3 and W = {(x, y, z) 2 R3 | x4 y 2 = 0}. (d) V = M 2⇥2 (R)◆be the vector space of 2 ⇥ 2 matrices with real coefficients and ⇢✓ a b W = 2 M2⇥2 (R) | 3a 2d = 0 . c d Ex. 12. Let W1 and W2 be subspaces of a vector space V . T (a) Prove that W1 W2 is a subspace S of V . (b) Give an example to show that W1 W2 need not be a subspace of V . (c) Is W1 \ W2 a subpace of V ? Ex. 13. Let S be a subset of a vector space V . Prove that Span(S) is a subspace of V . (Note: This is another common theorem that you could usually use without proof.) Ex. 14. Suppose that V is a vector space and u and v are vectors in V . Show that Span({u, v}) = Span({u + v, u v}). Ex. 15. Suppose that u, v, and w are vectors in a vector space V and that {u, v, w} is linearly independent. Prove that {u + 2v, v + 2w, u + 2w} is linearly independent. Ex. 16. Let V be a vector space. Suppose that {u, v, w} is a basis of V . Is the set {u + 2v, v + 2w, u + 2w} also a basis of V ? Justify your answer. Ex. 17. Let W = {(x, y, z) 2 R3 : x + 2y + 3z = 0}. (a) Prove that W is a subspace of R3 . (b) Find a basis of W . Ex. 18. Let U and V be subspaces of a vector space W . (a) Prove that U + V = {u + v : u 2 U, v 2 V } is a subspace of W . (b) Suppose {u1 , . . . um } is a basis for U and {v1 , . . . vn } is a basis for V . Prove that {u1 , . . . , um , v1 , . . . vn } spans U + V . (c) Prove that dim(U + V )  dim(U ) + dim(V ). (d) Give an example of a vector space W with subspaces U and V where dim(U + V ) < dim(U ) + dim(V ).