Answer to Question #195401 in Linear Algebra for cayyy

Question

Suppose U₁; U₂;...; U_m are finite-dimensional subspaces of V .

Prove that U₁ + U₂ +...+ U_m is finite-dimensional and dim(U₁ + U₂ +.....+ U_m) ≤ dim U₁ + dim U₂ +...+ dim U_m.

Answer 1

Proof by induction method on m.

Base case.

In the m = 1 case,

this formula reduces to

dim(U₁) ≤ dim(U₁),

which is trivial.

Inductive step.

In the m=m-1 case,

We assume that

dim(U₁ +···+U_m-1) ≤ dim(U₁)+···+dim(U_m-1) ..............eq(1)

and

we will prove that

dim(U₁ +··· +U_m-1 +U_m) ≤ dim(U₁)+···+dim(U_m-1)+dim(U_m).

Let W =U₁+···+U_m-1.

By a theorem in Axler and our inductive hypothesis,

dim(W +U_m)

=dim(W)+dim(U_m)−dim(W∩U_m)

≤dim(W)+dim(U_m)

≤(dim(U₁)+···+dim(U_m-1))+dim(U_m), as eq(1).

now put value of W, we get

dim(U₁ +··· +U_m-1 +U_m) ≤ dim(U₁)+···+dim(U_m-1)+dim(U_m).

Therefore, by the PMI, the inequality holds for every m≥1.