Linear Algebra

138 Chapter Two. Vector Spaces(b) Assume that S 1 is a linearly independent subset of W 1 and that S 2 is alinearly independent subset of W 2 . Can S 1 ∪S 2 be a linearly independent subsetof W 1 + W 2 ? Must it?4.35 When a vector space is decomposed as a direct sum, the dimensions of thesubspaces add to the dimension of the space. The situation with a space that isgiven as the sum of its subspaces is not as simple. This exercise considers thetwo-subspace special case.(a) For these subspaces of M 2×2 find W 1 ∩ W 2 , dim(W 1 ∩ W 2 ), W 1 + W 2 , anddim(W 1 + W 2).( ) ( )0 0 ∣∣ 0 b ∣∣W 1 = { c, d ∈ R} W2 = { b, c ∈ R}c dc 0(b) Suppose that U and W are subspaces of a vector space. Suppose that thesequence 〈 β ⃗ 1 , . . . , β ⃗ k 〉 is a basis for U ∩ W . Finally, suppose that the priorsequence has been expanded to give a sequence 〈⃗µ 1, . . . , ⃗µ j, β ⃗ 1, . . . , β ⃗ k 〉 that is abasis for U, and a sequence 〈 β ⃗ 1 , . . . , β ⃗ k , ⃗ω 1 , . . . , ⃗ω p 〉 that is a basis for W . Provethat this sequence〈⃗µ 1, . . . , ⃗µ j, β ⃗ 1, . . . , β ⃗ k , ⃗ω 1, . . . , ⃗ω p〉is a basis for for the sum U + W .(c) Conclude that dim(U + W ) = dim(U) + dim(W ) − dim(U ∩ W ).(d) Let W 1 and W 2 be eight-dimensional subspaces of a ten-dimensional space.List all values possible for dim(W 1 ∩ W 2 ).4.36 Let V = W 1 ⊕ . . . ⊕ W k and for each index i suppose that S i is a linearlyindependent subset of W i . Prove that the union of the S i ’s is linearly independent.4.37 A matrix is symmetric if for each pair of indices i and j, the i, j entry equalsthe j, i entry. A matrix is antisymmetric if each i, j entry is the negative of the j, ientry.(a) Give a symmetric 2×2 matrix and an antisymmetric 2×2 matrix. (Remark.For the second one, be careful about the entries on the diagional.)(b) What is the relationship between a square symmetric matrix and its transpose?Between a square antisymmetric matrix and its transpose?(c) Show that M n×n is the direct sum of the space of symmetric matrices andthe space of antisymmetric matrices.4.38 Let W 1 , W 2 , W 3 be subspaces of a vector space. Prove that (W 1 ∩W 2 )+(W 1 ∩W 3) ⊆ W 1 ∩ (W 2 + W 3). Does the inclusion reverse?4.39 The example of the x-axis and the y-axis in R 2 shows that W 1 ⊕ W 2 = V doesnot imply that W 1 ∪ W 2 = V . Can W 1 ⊕ W 2 = V and W 1 ∪ W 2 = V happen?̌ 4.40 Our model for complementary subspaces, the x-axis and the y-axis in R 2 ,has one property not used here. Where U is a subspace of R n we define theorthocomplement of U to beU ⊥ = {⃗v ∈ R n ∣ ∣ ⃗v ⃗u = 0 for all ⃗u ∈ U}(read “U perp”).(a) Find the orthocomplement of the x-axis in R 2 .(b) Find the orthocomplement of the x-axis in R 3 .(c) Find the orthocomplement of the xy-plane in R 3 .(d) Show that the orthocomplement of a subspace is a subspace.(e) Show that if W is the orthocomplement of U then U is the orthocomplementof W .