Results 1 to 3 of 3

Math Help - linear operator proof

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    8

    linear operator proof

    Let W be the space of all n X 1 column matrices over a field F. If A is an
    n X n matrix over F, then A defines a linear operator L on W through left
    multiplication: L(X) = AX. Prove that every linear operator on W is left multiplication
    by some n X n matrix, i.e., is L for some A.


    Now suppose V is an n-dimensional vector space over the field F, and let b be an ordered basis for V. For each a in V, define Ua = [a]b (a in b-coordinates). Prove that U is an isomorphism of V onto W. If T is a linear operator on V, then UTU^(-1) is a linear operator on W. Accordingly, UTU^(-1) is left multiplication by some n X n matrix A. What is A?

    Any help on these two problems would be appreciated.
    Last edited by guroten; October 11th 2009 at 12:42 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,444
    Thanks
    1863
    Quote Originally Posted by guroten View Post
    Let W be the space of all n X 1 column matrices over a field F. If A is an
    n X n matrix over F, then A defines a linear operator L on W through left
    multiplication: L(X) = AX. Prove that every linear operator on W is left multiplication
    by some n X n matrix, i.e., is L for some A.
    Choose a basis for W, [tex]\{M_1, M_2, ..., M_n}. Taking the column matrices with exactly one "1" and the rest of the entries "0", so that M_1= \begin{bmatrix}1 \\ 0 \\ 0\\...\\0\end{bmatrix}, M_2=\begin{bmatrix}0 \\ 1 \\ 0\\...\\0\end{bmatrix} will work nicely. For linear transformation L, write L(M_1) as a linear combination of the the basis. That coeffientes in that linear combination will give you the first column of A. Continue with L(M_2), etc. to find the other columns.

    This is a general method. If L is a linear transformation from vector space U to vector space V, choose bases for U and V. Apply L to each basis vector of U in turn and write the resulting vector (in V) as a linear combination of the chosen basis for V. The coefficients will give the columns of the matrix representation of L.

    Now suppose V is an n-dimensional vector space over the field F, and let a
    be an ordered basis for V. For each a in V, define Ua = [a]b. Prove that U is an
    isomorphism of V onto W.
    You've lost me here. You say "let a be an ordered basis" but then say "for each a in V". Did you mean "let b be an ordered basis"? Does "[a]b" mean "the coefficients when a is written as a linear combination in that basis"? (I was try to think of "b" as a vector!) "W" is still the column matrices, right? Remember the basis I suggested for W above? Do you see that U maps each vector of b into the corresponding member of that basis. And any linear transformation that maps basis vector to basis vectors is an isomorphism.

    [quote]
    If T is a linear operator on V, then UTU^(-1) is a linear
    operator on W. Accordingly, UTU^(-1) is left multiplication by some n X n matrix A.
    What is A?
    Basically, what I said before. A is the matrix representing T in this basis. Its columns are the coefficients in the linear combination of each T(v_i) in the basis {v_i\}.

    Any help on these two problems would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2009
    Posts
    8
    Sorry, I messed up the second part. It should read:


    Now suppose V is an n-dimensional vector space over the field F, and let b
    be an ordered basis for V. For each a in V, define Ua = [a]b (a in b-coordinates). Prove that U is an isomorphism of V onto W. If T is a linear operator on V, then UTU^(-1) is a linear operator on W. Accordingly, UTU^(-1) is left multiplication by some n X n matrix A. What is A?

    I think that corrects it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear operator
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: June 5th 2011, 11:41 PM
  2. Linear operator proof
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 20th 2011, 11:30 PM
  3. Linear operator proof
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: March 27th 2010, 05:16 AM
  4. linear operator
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 13th 2008, 09:24 PM
  5. Proof for Isomorphic Linear Operator
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 3rd 2007, 06:14 PM

Search Tags


/mathhelpforum @mathhelpforum