Results 1 to 2 of 2

Thread: Help with change of basis proof

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    9

    Help with change of basis proof

    Let V be a finite-dimensional vector space over F, and suppose that T ∈ L(V ) is a linear operator having the following property: Given any two bases b and c for V , the matrix M(T, b) for T with respect to b is the same as the matric M(T, c) for T with respect to c. Prove that there exists a scalar α ∈ F such that T = αidV , where idV denotes the identity map on V .

    I don't really know how to approach this, and some guidance would be helpful! Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by blackwrx View Post
    Let V be a finite-dimensional vector space over F, and suppose that T ∈ L(V ) is a linear operator having the following property: Given any two bases b and c for V , the matrix M(T, b) for T with respect to b is the same as the matric M(T, c) for T with respect to c. Prove that there exists a scalar α ∈ F such that T = αidV , where idV denotes the identity map on V .

    I don't really know how to approach this, and some guidance would be helpful! Thanks.


    Let's see if you can follow me: any element in $\displaystyle L(V)$ can be seen as a change of basis transformation, since a linear map is invertible iff it maps a basis into a basis, so what the problem is giving is that T is a linear map s. t. $\displaystyle STS^{-1}=T\Longleftrightarrow ST =TS\,,\,\,\forall \,S\in L(V)$ (why?).

    What's left now is to prove that the only lin. maps that commute with ALL the elements in L(V) are the scalar multiples of the identity map, and this is a nice, fairly non-hard exercise. One way to approach it is to work with matrices...

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Change of basis
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Mar 14th 2011, 04:45 AM
  2. Change of Basis Help
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Feb 21st 2011, 04:47 AM
  3. Change of basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Aug 29th 2010, 03:08 PM
  4. Change of basis?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Mar 12th 2010, 04:30 AM
  5. Change of Basis- Please help
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 18th 2009, 12:46 AM

Search Tags


/mathhelpforum @mathhelpforum