I just need some guidance in starting these proofs, i.e. where to start with them.
1. Let V be a finite-dimensional inner product space over F, and let U be a subspace of V . Prove that the orthogonal complement U⊥ of U with respect to the inner product <· , ·> on V satisfies
dim(U⊥) = dim(V ) − dim(U).
2. Let V be a finite-dimensional inner product space over F, and suppose that P ∈ L(V ) is a linear operator on V having the following two properties:
(a) Given any vector v ∈ V , P(P(v)) = P(v). I.e., P2 = P.
(b) Given any vector u ∈ null(P) and any vector v ∈ range(P), <u, v> = 0.
Prove that P is an orthogonal projection.
3. Prove or give a counterexample: For any n ≥ 1 and A ∈ Cn×n, one has
null(A) = (range(A))⊥.
I am fairly new to proof writing and am just looking on some ideas to help
Thanks in advance.