For every vector w ∈ R 3 , Aw is in the span of u. Prove this.I am unsure how to approach this question. Given is A = ( 2 − 1 3 4 − 2 6 − 2 1 − 3 ) and u = ( 1 2 − 1 ) .We have to prove for every w ∈ R 3 that Aw is in the span of u. So far I have calculated and written Aw as A w = ( 2 w 1 − w 2 + 3 w 3 4 w 1 − 2 w 2 + 6 w 3 − 2 w 1 + w 2 − 3 w 3 ) And I know that s p ( u ) = { r 1 + 2 r 2 − r 3 ∣ r 1 , r 2 , r 3 ∈ R }But I do not really know where to go from here. Any help or guidance is appreciated.

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For every vector   w  ∈            R        3  , Aw  is in the span of u. Prove this.I am unsure how to approach this question. Given is  A  =      (                            2                          −          1                          3                                      4                          −          2                          6                                      −          2                          1                          −          3                      )  and  u  =      (                            1                                      2                                      −          1                      )    .We have to prove for every   w  ∈            R        3   that Aw is in the span of u. So far I have calculated and written Aw as  A  w  =      (                            2                      w            1                    −                      w            2                    +          3                      w            3                                                4                      w            1                    −          2                      w            2                    +          6                      w            3                                                −          2                      w            1                    +                      w            2                    −          3                      w            3                                )  And I know that   s  p  (  u  )  =  {      r    1    +  2      r    2    −      r    3    ∣      r    1    ,      r    2    ,      r    3    ∈      R    }But I do not really know where to go from here. Any help or guidance is appreciated.

reinmelk3iu · Accepted Answer

A  w  =      (                            2                      w            1                    −                      w            2                    +          3                      w            3                                                4                      w            1                    −          2                      w            2                    +          6                      w            3                                                −          2                      w            1                    +                      w            2                    −          3                      w            3                                )    =  (  2      w    1    −      w    2    +  3      w    3    )      (                            1                                      2                                      −          1                      )  Recall that the span of a set of vectors is the set of linear combinations of those vectors.

MISA6zh · Accepted Answer

The column space, upon perusal, is the span of   (  1  ,  2  ,  −  1  ). But in general, given a linear transformation represented by a matrix (rel the standard basis), the image is the column space.

For every vector w in RR^3, Aw is in the span of u. Prove this.

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