Linear Algebra

134 Chapter Two. Vector SpacesIn this subsection we have seen two ways to regard a space as built up fromcomponent parts. Both are useful; in particular we will use the direct sumdefinition to do the Jordan Form construction at the end of the fifth chapter.Exerciseš 4.20 Decide if(R) 2 is the direct sum(of)each W 1 and W 2 .x ∣∣ x ∣∣(a) W 1 = { x ∈ R}, W2 = { x ∈ R}0x( ) ( ) s ∣∣ s ∣∣(b) W 1 = { s ∈ R}, W2 = { s ∈ R}s1.1s(c) W 1 = R 2 , W 2 = {⃗0}( ) t ∣∣(d) W 1 = W 2 = { t ∈ R}t( ( ) ( ( ) 1 x ∣∣ −1 0 ∣∣(e) W 1 = { + x ∈ R}, W2 = { + y ∈ R}0)00)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⎛ ⎞z{ ⎝z⎠ ∣ z ∈ R}z4.22 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 pairwiseintersections be trivial is genuinely stronger than the requirement only that theintersection 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,the intersection of all three subspaces W 1 ∩ W 2 ∩ W 3 is trivial, but the pairwiseintersections W 1 ∩ W 2 , W 1 ∩ W 3 , and W 2 ∩ W 3 are nontrivial.