Linear Algebra

136 Chapter Two. Vector Spaces4.41 We know that if V = W 1 ⊕ W 2 then there is a basis for V that splits into abasis for W 1 and a basis for W 2 . Can we make the stronger statement that everybasis for V splits into a basis for W 1 and a basis for W 2 ?4.42 We can ask about the algebra of the ‘+’ operation.(a) Is it commutative; is W 1 + W 2 = W 2 + W 1 ?(b) Is it associative; is (W 1 + W 2 ) + W 3 = W 1 + (W 2 + W 3 )?(c) Let W be a subspace of some vector space. Show that W + W = W.(d) Must there be an identity element, a subspace I such that I + W = W + I = Wfor all subspaces W?(e) Does left-cancellation hold: if W 1 + W 2 = W 1 + W 3 then W 2 = W 3 ? Rightcancellation?4.43 Consider the algebraic properties of the direct sum operation.(a) Does direct sum commute: does V = W 1 ⊕ W 2 imply that V = W 2 ⊕ W 1 ?(b) Prove that direct sum is associative: (W 1 ⊕ W 2 ) ⊕ W 3 = W 1 ⊕ (W 2 ⊕ W 3 ).(c) Show that R 3 is the direct sum of the three axes (the relevance here is that bythe previous item, we needn’t specify which two of the three axes are combinedfirst).(d) Does the direct sum operation left-cancel: does W 1 ⊕ W 2 = W 1 ⊕ W 3 implyW 2 = W 3 ? Does it right-cancel?(e) There is an identity element with respect to this operation. Find it.(f) Do some, or all, subspaces have inverses with respect to this operation: isthere a subspace W of some vector space such that there is a subspace U withthe property that U ⊕ W equals the identity element from the prior item?