# Thread: Inner Product Proof Help

1. ## Inner Product Proof Help

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

2. Originally Posted by blackwrx
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).

Take basis for $U\,,\,\,U^{\perp}$ and show that their set theoretical union is a basis for the whole space

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.

Definition of orthogonal projection

3. Prove or give a counterexample: For any n ≥ 1 and A ∈ Cn×n, one has

null(A) = (range(A))⊥.

What is Cnxn , anyway?

Tonio

I am fairly new to proof writing and am just looking on some ideas to help

.

3. Originally Posted by blackwrx
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).
Let $\{u_1, u_2, \cdot\cdot\cdot, u_m\}$ be a basis for U. Let $\{w_1, w_2, \cdot\cdot\cdot, w_i\}$ be a basis for U⊥.

Show that $\{u_1, u_2, \cdot\cdot\cdot, u_m,w_1, w_2, \cdot\cdot\cdot, w_i\}$ is a basis for V.

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.
What is the definition of "orthogonal projection"?
What can you say about the set $\{P(v)| v\in V\}$?

3. Prove or give a counterexample: For any n ≥ 1 and A ∈ Cn×n, one has

null(A) = (range(A))⊥.
How you would prove this depends upon what you have available. If you can use the "rank-nullity" theorem, that if $A:U\rightarrow V$ then nullity(A)+ rank(A)= dim(U) it is fairly straightforward.

I am fairly new to proof writing and am just looking on some ideas to help