Page 32 - Linear Algebra Done Right
P. 32
Chapter 1. Vector Spaces
18
v ∈ U and −v ∈ W. By the unique representation of 0 as the sum of a
vector in U and a vector in W, we must have v = 0. Thus U ∩ W ={0},
completing the proof in one direction.
To prove the other direction, now suppose that V = U + W and
U ∩ W ={0}. To prove that V = U ⊕ W, suppose that
0 = u + w,
where u ∈ U and w ∈ W. To complete the proof, we need only show
that u = w = 0 (by 1.8). The equation above implies that u =−w ∈ W.
Thus u ∈ U ∩ W, and hence u = 0. This, along with equation above,
implies that w = 0, completing the proof.
