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Math Help - Inner Product Proof Help

  1. #1
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    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 ∈ Cnn, 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.
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  2. #2
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    Quote Originally Posted by blackwrx View Post
    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 ∈ Cnn, 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

    Thanks in advance.
    .
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  3. #3
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    Quote Originally Posted by blackwrx View Post
    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 ∈ Cnn, 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

    Thanks in advance.
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  4. #4
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    Thanks guys!
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