Linear Algebra, 2020a

152 Chapter Two. Vector Spaces
4.39 The example of the x-axis and the y-axis in R 2 shows that W 1 ⊕ W 2 = V does
not imply that W 1 ∪ W 2 = V. Can W 1 ⊕ W 2 = V and W 1 ∪ W 2 = V happen?
4.40 Consider Corollary 4.13. Does it work both ways — that is, supposing that V =
W 1 +···+W k ,isV = W 1 ⊕· · ·⊕W k if and only if dim(V) =dim(W 1 )+···+dim(W k )?
4.41 We know that if V = W 1 ⊕ W 2 then there is a basis for V that splits into a
basis for W 1 and a basis for W 2 . Can we make the stronger statement that every
basis 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 = W
for all subspaces W?
(e) Does left-cancellation hold: if W 1 + W 2 = W 1 + W 3 then W 2 = W 3 ? Right
cancellation?