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Linear Transformation

  1. May 2, 2007 #1
    1) True or False? If true, prove it. If false, prove that it is false or give a counterexample.
    1a) If a linear transformation T: R^n->R^m is onto and R^n = span{X1,...,Xk}, then R^m = span{T(X1),...,T(Xk)}
    1b) If T: R^n->R^m is a linear transformation and U is a subspace of R^n, then T(U) is a subspace of R^m.




    2) Let T: R^2->R^4 be a linear transformation induced by the matrix A=
    [1 4
    2 3
    3 2
    4 1]
    Find a vector X E R^2 such that T(X) is as close as possible to [4 6 6 4]^T




    I have an exam tomorrow. These are the past exams questions that I am having terrible trouble with. Can someone help me? I seriously thought about these questions, but still can't come up with any clue...I really want to provide some attempt, but I don't even know how to begin...

    Any help/hints is greatly appreciated!
     
  2. jcsd
  3. May 2, 2007 #2
    1 (a) First what is the definition of onto or surjective? Every point in the domain spans the codomain. Use this fact to answer your question.
     
    Last edited: May 2, 2007
  4. May 2, 2007 #3

    radou

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    Regarding 1b), simply take two vectors a, b from T(U) and think of a condition which must be satisfied in order for T(U) to be a subspace.
     
  5. May 2, 2007 #4

    siddharth

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    Gold Member



    For this question, you need to find a "best approximation" [tex] u \in \mathbb{R}^2[/tex] to
    [tex]b= \left( \begin{array}{c} 4 \\ 6 \\6 \\ 4 \end{array} \right)[/tex]

    Have you learnt the theorem which says that if u is a best approximation, and A is the matrix of the linear transformation, [tex]A^T(Au-b)=0[/tex]?

    Solve for u to find the best approximation. You could apply QR factorization to A to further simply the solution process.
     
    Last edited: May 2, 2007
  6. May 2, 2007 #5
    Thanks, I have learnt this but I have never thought of it...what a great method
     
    Last edited: May 2, 2007
  7. May 2, 2007 #6
    But I am still pretty lost with question 1b...

    I know the definition of subspace, but I simply don't know how to apply it in this situation...

    U is a subsapce of V iff
    1) 0 E U
    2) X,Y E U => X+Y E U
    and 3) X E U, a E R => aX E U
     
  8. May 2, 2007 #7

    radou

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    U is a subspace of V if, for every a, b from U, and for every salars x, y, xa+yb is in U. Use that fact.
     
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