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Linear algebra determinant of linear operator

  1. Mar 26, 2008 #1
    [SOLVED] linear algebra determinant of linear operator

    1. The problem statement, all variables and given/known data

    Let T be a linear operator on a finite-dimensional vector space V.
    Define the determinant of T as: det(T)=det([T]β) where β is any ordered basis for V.

    Prove that for any scalar λ and any ordered basis β for V that det(T - λIv) = det([T]β - λI).

    2. Relevant equations

    Another part of the problem yielded that for any two ordered bases of V, β and γ , that det([T]β) = det([T]γ).


    3. The attempt at a solution

    I need someone to help me understand the notation. I don't actually know what I am being asked to prove here.
     
  2. jcsd
  3. Mar 26, 2008 #2

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    T is the general linear transformation. [T] is the matrix that represents the linear transformation with respect to the basis B. If we change the basis, then we change the matrix since if TX = Y where X and Y are the coordinate vectors in K^n representing the abstract vectors x,y in V. (We can do with since any n dimensional vector space is isomorphic to K^n.) These coordinates are just the coefficients of the basis vectors.
    ie:if x = a1v1 +a2v2 + .. + anvn where v1 ...vn is a basis then the coordinate vector for x just (a1,a2,a3,...,an). Obviously these coordinates change when the basis changes. The same holds for Y which means our matrix will need to change to reflect this.

    Also, X and Y are column vectors so that the matrix multiplication is defined.
     
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