Let { x , y , z } be the basis of R 3 and A : R 3 → R 3 is a linear operator given by the matrix: [ − 1 − 2 − 5 0 2 2 2 2 4 ] a) Determine the transformation matrix A with respect to the basis { x + y , y + z , z + x } .b)Determine if A is a linear transformation. AWhat is the process in determining the transformation matrix with respect to a new basis when you already have a transformation matrix with respect to another?

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Let   {  x  ,  y  ,  z  }  be the basis of         R    3   and   A  :        R    3    →        R    3   is a linear operator given by the matrix:      [                            −          1                          −          2                          −          5                                      0                          2                          2                                      2                          2                          4                      ]  a) Determine the transformation matrix   A with respect to the basis   {  x  +  y  ,  y  +  z  ,  z  +  x  }  .b)Determine if A is a linear transformation.  AWhat is the process in determining the transformation matrix with respect to a new basis when you already have a transformation matrix with respect to another?

luisjoseblash2 · Accepted Answer

For question 2,   A is a linear transformation because it can be represented by a matrix.For question 1, first of all, we have  A      x    →    =  −      x    →    +  2      z    →      A      y    →    =  −  2      x    →    +  2      y    →    +  2      z    →      A      z    →    =  −  5      x    →    +  2      y    →    +  4      z    →  You can see that the columns of   A are the coefficients of the vector obtained when you apply   A to the basis.Same idea leads to the matrix under another basis. We have  A  (      x    →    +      y    →    )  =  −  3      x    →    +  2      y    →    +  4      z    →  using the above equations.Now write   −  3      x    →    +  2      y    →    +  4      z    →   in terms of       x    →    +      y    →    ,      y    →    +      z    →    ,      z    →    +      x    →    , by some computation  −  3      x    →    +  2      y    →    +  4      z    →    =  −      5    2    (      x    →    +      y    →    )  +      9    2    (      y    →    +      z    →    )  −      1    2    (      z    →    +      x    →    )So the first column of the new matrix should be      (                            −                      5            2                                                            9            2                                                −                      1            2                                )  You can find the other columns using similar way.

Let <mo fence="false" stretchy="false">{ x , y , z <mo fence="false" stretchy="false"

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