Question

Let W1 and W2 be 2 subspaces of a finite dimensional vector space V. If dim V = dim W1 + dim W2 and W1⋂W2 = {0} prove that V = W1 ⊕ W2

Answer 1

Step-by-step explanation:

One direction is easy : Let α≠0 and α∈W

⊥

1

+W

⊥

2

, i.e. α can be written as α=β+γ such that β∈W

⊥

1

and γ∈W

⊥

2

, hence (β|η)=0 for all η∈W1 and (γ|δ)=0 for all δ∈W2. Now for all η∈W1∩W2 it is clear that (α|η)=0 . hence

(W1∩W2)⊥⊃W

⊥

1

+W

⊥

2

For proving the other containment, let α≠0 and α∈(W1∩W2)⊥, it means that for all β∈W1∩W2, (α|β)=0. Hence α∈V∖(W1∩W2)=V∖(W1)∪V∖(W2). Hence α∈W

c

1

or α∈W

c

2

. WLOG suppose α∈W

c

1

. We also have that

V=(W1)⊕(W1)⊥

therefore α=η+δ where η∈W1 and δ∈W

⊥

1

and indeed δ≠0. From here I have to somehow show that η=0, but I am stuck in here...