Let there be a linear transformation going from R 3 to R 2 , defined by T ( x , y , z ) = ( x + y , 2 z − x ). Find the transformation matrix if base 1: ⟨ ( 1 , 0 , − 1 ) , ( 0 , 1 , 1 ) , ( 1 , 0 , 0 ) ⟩, base 2: ⟨ ( 0 , 1 ) , ( 1 , 1 ) ⟩An attempt at a solution included calculating the transformation on each of the bases in R 3 , (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of B 1 in B 2 Another point: if the basis for R 3 and R 2 are the standard basis for these spaces, the attempt at a solution is a correct answer.

Question

Let there be a linear transformation going from         R    3   to         R    2  , defined by   T  (  x  ,  y  ,  z  )  =  (  x  +  y  ,  2  z  −  x  ). Find the transformation matrix if base 1:  ⟨  (  1  ,  0  ,  −  1  )  ,  (  0  ,  1  ,  1  )  ,  (  1  ,  0  ,  0  )  ⟩, base 2:   ⟨  (  0  ,  1  )  ,  (  1  ,  1  )  ⟩An attempt at a solution included calculating the transformation on each of the bases in         R    3  , (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of       B    1   in       B    2  Another point: if the basis for         R    3   and         R    2    are the standard basis for these spaces, the attempt at a solution is a correct answer.

Hadley Cunningham · Accepted Answer

Let us call the basis of         R      3     {      λ      1    ,      λ      2    ,      λ      3    } and the basis of         R      2     {      γ      1    ,      γ      2    }So all you need to do now is the following:  T  (      λ      1    )  =      α      11        γ      1    +      α      21        γ      1    T  (      λ      2    )  =      α      12        γ      1    +      α      22        γ      1    T  (      λ      3    )  =      α      13        γ      1    +      α      23        γ      1  while   ∀      α          i      ,      j        ∈  RAnd the matrix will look like:  [  T  ]  =  (                              α                  11                                      α                  12                                      α                  13                                              α                  21                                      α                  22                                      α                  23                      )And this is how it is done

Let there be a linear transformation going from <mi mathvariant="double-struck">R 3

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