Linear Algebra

Section III. Basis and Dimension 137( )t ∣∣(d) W 1 = W 2 = { t ∈ R}t( (1 x(e) W 1 = { +0)0) ∣∣x ∈ R}, W2 = {(−10) ( )0 ∣∣+ y ∈ R}y̌ 4.21 Show that R 3 is the direct sum of the xy-plane with each of these.(a) the z-axis(b) the line{( zzz)∣∣z ∈ R}∣ ∣4.22 Is P 2 the direct sum of {a + bx 2 a, b ∈ R} and {cx c ∈ R}?̌ 4.23 In P n, the even polynomials are∣the members of this setE = {p ∈ P n p(−x) = p(x) for all x}and the odd polynomials are the members∣of this set.O = {p ∈ P n p(−x) = −p(x) for all x}Show that these are complementary subspaces.4.24 Which of these subspaces of R 3W 1: the x-axis, W 2: the y-axis, W 3: the z-axis,W 4 : the plane x + y + z = 0, W 5 : the yz-planecan be combined to(a) sum to R 3 ? (b)∣direct sum to R 3 ?∣̌ 4.25 Show that P n = {a 0 a0 ∈ R} ⊕ . . . ⊕ {a n x n an ∈ R}.4.26 What is W 1 + W 2 if W 1 ⊆ W 2 ?4.27 Does Example 4.5 generalize? That is, is this true or false: if a vector space Vhas a basis 〈 β ⃗ 1, . . . , β ⃗ n〉 then it is the direct sum of the spans of the one-dimensionalsubspaces V = [{ β ⃗ 1 }] ⊕ . . . ⊕ [{ β ⃗ n }]?4.28 Can R 4 be decomposed as a direct sum in two different ways? Can R 1 ?4.29 This exercise makes the notation of writing ‘+’ between sets more natural.Prove that, where W 1, . . . , W k are subspaces of∣a vector space,W 1 + · · · + W k = { ⃗w 1 + ⃗w 2 + · · · + ⃗w k ⃗w1 ∈ W 1 , . . . , ⃗w k ∈ W k },and so the sum of subspaces is the subspace of all sums.4.30 (Refer to Example 4.19. This exercise shows that the requirement that pariwiseintersections be trivial is genuinely stronger than the requirement only thatthe intersection of all of the subspaces be trivial.) Give a vector space and threesubspaces W 1 , W 2 , and W 3 such that the space is the sum of the subspaces, theintersection of all three subspaces W 1 ∩ W 2 ∩ W 3 is trivial, but the pairwise intersectionsW 1 ∩ W 2 , W 1 ∩ W 3 , and W 2 ∩ W 3 are nontrivial.̌ 4.31 Prove that if V = W 1 ⊕ . . . ⊕ W k then W i ∩ W j is trivial whenever i ≠ j. Thisshows that the first half of the proof of Lemma 4.15 extends to the case of morethan two subspaces. (Example 4.19 shows that this implication does not reverse;the other half does not extend.)4.32 Recall that no linearly independent set contains the zero vector. Can anindependent set of subspaces contain the trivial subspace?̌ 4.33 Does every subspace have a complement?̌ 4.34 Let W 1 , W 2 be subspaces of a vector space.(a) Assume that the set S 1 spans W 1, and that the set S 2 spans W 2. Can S 1 ∪S 2span W 1 + W 2 ? Must it?