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Change of Basis between different size spaces

  1. Feb 29, 2012 #1
    Hi,

    I'm working on an example question with the following info:

    [itex]\alpha[/itex] = {(3,0,1) , (3,1,1), (2,1,1)} [itex]\beta[/itex] = {(1,1), (1,-1)} Are a set of bases. [T][itex]\beta\alpha[/itex] = \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1 \end{bmatrix} Now they go on to say:

    Let T: R3--> R3 be the transformation whose matris with respect to the basis [itex]\alpha[/itex] is:


    [T][itex]\\alpha\alpha[/itex] = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}

    Now I'm trying to do some back calculations to figure out how they got that matrix, but the only thing I know would be to use a calculation such as [itex]\beta\alpha[/itex] [T][itex]\alpha\beta[/itex]


    But I can't write out a vector in R2 as a linear combination of vectors in R3 right? So how would I get that

    [T][itex]\\alpha\alpha[/itex] = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}
     
  2. jcsd
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