Given a matrix A = [ 1 1 1 0 1 1 0 0 1 ] and bases to a the vector space V: B = ( v 1 , v 2 , v 3 ) , B 1 = ( v 1 , v 1 + v 2 , v 1 + v 2 + v 3 )Is A the transformation matrix between the basis B to B 1 ?

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Given a matrix   A  =      [                            1                          1                          1                                      0                          1                          1                                      0                          0                          1                      ]   and bases to a the vector space   V:  B  =  (      v      1    ,      v      2    ,      v      3    )  ,        B      1    =  (      v      1    ,      v      1    +      v      2    ,      v      1    +      v      2    +      v      3    )Is A the transformation matrix between the basis   B to       B    1  ?

Korotnokby · Accepted Answer

Let       [                            0                                      1                                      0                      ]  ,       [                            0                                      1                                      0                      ]  , and       [                            0                                      0                                      1                      ]   be the coordinate vectors representing       v    1  ,       v    2   and       v    3  respectively with respect to the basis   B. Then we can see that the transformation   x  ↦  A  x maps each of these three vectors to the coordinates of       v    1    ,         v    1    +      v    2    , and       v    1    +      v    2    +      v    3   with respect to the   {      v    1    ,      v    2    ,      v    3    } basis. Then we see how these coordinate vectors transform:   A  =      [                            1                          1                          1                                      0                          1                          1                                      0                          0                          1                      ]        [                            1                                      0                                      0                      ]    =      [                            1                                      0                                      0                      ]  and  A  =      [                            1                          1                          1                                      0                          1                          1                                      0                          0                          1                      ]        [                            0                                      1                                      0                      ]    =      [                            1                                      1                                      0                      ]  and  A  =      [                            1                          1                          1                                      0                          1                          1                                      0                          0                          1                      ]        [                            0                                      0                                      1                      ]    =      [                            1                                      1                                      1                      ]  Notice that this is just the coordinate vectors of       v    1  ,       v    1    +      v    2  , and       v    1    +      v    2    +      v    3   with respect to the basis   B.So under this interpretation, the mapping   x  ↦  A  x transforms the vector   x itself without changing the basis

veirarer · Accepted Answer

Let       [                            1                                      0                                      0                      ]  ,       [                            0                                      1                                      0                      ]  , and       [                            0                                      0                                      1                      ]   be the coordinate vectors representing       v    1  ,       v    1    +      v    2  , and       v    1    +      v    2    +      v    3  , respectively, with respect to the basis       B    1  So in this case we can think of this transformation as mapping vectors with respect to the       B    1   basis to their coordinates in the basis   BUnder this interpretation, your mapping is actually the opposite of what you thought -- it maps vectors from the       B    1   basis to their coordinates w.r.t.   B basis. The one which maps vectors from   B to       B    1   would be the inverse of this matrix.

Given a matrix A = [ <mtable rowspacing="4pt" columnspacing="1em"> <mtr>

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