Let T ( x , y ) = ( x + 2 y , x − y ) and U ( x , y ) = ( x − y , 3 x + y ) be linear transformations from R 2 to R 2 Determine the transformations of the composition of U with T ( U ° T )So I can determine the standard matrices are T = [ 1 2 1 − 1 ] and U = [ 1 − 1 3 1 ] but I'm not sure how to use these with compositions. What is the process needed to answer the question?

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Let   T  (  x  ,  y  )  =  (  x  +  2  y  ,  x  −  y  ) and   U  (  x  ,  y  )  =  (  x  −  y  ,  3  x  +  y  ) be linear transformations from       R    2   to       R    2   Determine the transformations of the composition of U with   T   (  U      °    T  )So I can determine the standard matrices are   T  =      [                            1                          2                                      1                          −          1                      ]   and   U  =       [                            1                          −          1                                      3                          1                      ]   but I&#039;m not sure how to use these with compositions. What is the process needed to answer the question?

Marshall Mcpherson · Accepted Answer

Forget about matrices. Composition is defined more generally, and in this case it is pretty simple to work out what it is: for every   (  x  ,  y  )  ∈            R        2  , we have                    (        U        ∘        T        )        (        x        ,        y        )                            :=        U        (        T        (        x        ,        y        )        )                                          =        U        (        x        +        2        y        ,        x        −        y        )                                          =                              (                          (        x        +        2        y        )        −        (        x        −        y        )        ,        3        (        x        +        2        y        )        +        (        x        −        y        )                              )                                                            =        (        3        y        ,        4        x        +        5        y        )            Composition just means you do one function after the other, and since you have explicit formulas for both functions, just plug them in directly (and carefully).If you really want to think in terms of matrices (which I think is counterproductive here), then the matrix representation of   U  ∘  T is the product of the matrix representations of U and T (all matrices calculated relative to standard bases); i.e                              (                      ∗                    )                            [        U        ∘        T        ]                            =        [        U        ]        ⋅        [        T        ]                                          =                  (                                                    1                                            −                1                                                                    3                                            1                                              )                ⋅                  (                                                    1                                            2                                                                    1                                            −                1                                              )                                                  =                  (                                                    0                                            3                                                                    4                                            5                                              )                    If you revert back to function notation, this says   (  U  ∘  T  )  (  x  ,  y  )  =  (  3  y  ,  4  x  +  5  y  ). Notice how this is a three step-process (which is completely unnecessary); you first have to write down the matrices of U and T, then you have to multiply them, and then you have to &quot;undo&quot; the matrix of the product to figure out   U  ∘  TMoreover, from a conceptual standpoint, you should think of matrix multiplication as being motivated by function composition; that&#039;s really why   (  ∗  ) is true (matrix multiplication is defined in that funny way of multiplying rows and columns with the sole intention of making   (  ∗  ) work out).

Let T(x,y)=(x+2y,x−y) and U(x,y)=(x−y,3x+y) be linear transformations from R^2 to R^2 Determine the transformations of the composition of U with T (U^(circ) T)

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