Consider the transformation T : R 3 → R 2 defined by: T ( x ) = T ( x 1 x 2 x 3 ) = ( 2 x 1 + x 3 ) ( 1 2 ) + ( x 2 − 3 x 3 ) ( − 1 1 ) 1a) determine the matrix of the above transformation1b) determine the reduced row echelon form of the matrix found in 1a 1c) based on your answer to part 1b, is the transformation T onto?1d) based on your answer to part 1b, is the transformation T one-to-one?1e) based on your answer to part 1b, determine the set of vectors x in R 3 for which T ( x ) = 0. Write your answer in parametric vector form

Question

Consider the transformation   T  :        R    3    →        R    2   defined by:  T  (  x  )  =  T      (                                        x            1                                                            x            2                                                            x            3                                )    =  (  2      x    1    +      x    3    )      (                            1                                      2                      )    +  (      x    2    −  3      x    3    )      (                            −          1                                      1                      )  1a) determine the matrix of the above transformation1b) determine the reduced row echelon form of the matrix found in 1a 1c) based on your answer to part 1b, is the transformation T onto?1d) based on your answer to part 1b, is the transformation T one-to-one?1e) based on your answer to part 1b, determine the set of vectors   x in             R        3   for which   T  (  x  )  =  0. Write your answer in parametric vector form

postojahob · Accepted Answer

For part (a), notice that  T  (  x  )  =  (  2      x    1    +      x    3    )      [                            1                                      2                      ]    +  (      x    2    −  3      x    3    )      [                            −          1                                      1                      ]    =      [                            2                      x            1                    +                      x            3                                                4                      x            1                    +          2                      x            3                                ]    +      [                            −                      x            2                    +          3                      x            3                                                            x            2                    −          3                      x            3                                ]    =      [                            2                      x            1                    −                      x            2                    +          4                      x            3                                                4                      x            1                    +                      x            2                    −                      x            3                                ]    =      [                            2                          −          1                          4                                      4                          1                          −          1                      ]        [                                        x            1                                                            x            2                                                            x            3                                ]  Which implies that the matrix defining   T  (  x  ) is      [                            2                          −          1                          4                                      4                          1                          −          1                      ]  For 1e) I&#039;m getting the reduced-form matrix as :      [                            2                          −          1                          4                                      4                          1                          −          1                      ]    →      [                            1                          0                                      1            2                                                0                          1                          −          3                      ]  Which gives the system of equations      x    1    +      1    2        x    3    =  0      x    2    −  3      x    3    =  0Which implies that      x    1    =  −      1    2        x    3        x    2    =  3      x    3  Lastly, if we define       x    3    =  t, to solution in parametric form is then      x    1    =  −      1    2    t      x    2    =  3  t      x    3    =  t

fythynwyrk0 · Accepted Answer

For part a), it helps to rewrite  T  (  x  )  =      (                            2                      x            1                    −                      x            2                    +          4                      x            3                                                4                      x            1                    +                      x            2                    −                      x            3                                )  Now, for which matrix   A do we have   T  (  x  )  =  A  x?If you go through the rest of the problem correctly, you should find that the transformation is onto, but not one-to-one. Solving part e) is the same thing as finding the nullspace of the matrix.

Consider the transformation T : <mi mathvariant="double-struck">R 3 </msup

Answered question

Answer & Explanation

New Questions in Linear algebra