Find the coordinates of w = ( 2 , 5 , 1 ) T with respect to the basis B = { ( 1 , 0 , 0 ) T , ( 1 , 1 , 0 ) T , ( 1 , 1 , 1 ) T }Enter coordinates into the answer box below, separated with commas.

Question

Find the coordinates of   w  =  (  2  ,  5  ,  1      )    T   with respect to the basis  B  =  {  (  1  ,  0  ,  0      )    T    ,  (  1  ,  1  ,  0      )    T    ,  (  1  ,  1  ,  1      )    T    }Enter coordinates into the answer box below, separated with commas.

Dalton Erickson · Accepted Answer

Given that:  w  =  (  2  ,  5  ,  1      )    T     and    B  =  {  (  1  ,  0  ,  0      )    T    ,  (  1  ,  1  ,  0      )    T    ,  (  1  ,  1  ,  1      )    T    }    Let   (  2  ,  5  ,  1      )    T    =      c    1    (  1  ,  0  ,  0      )    T    +      c    2    (  1  ,  1  ,  0      )    T    +      c    3    (  1  ,  1  ,  1      )    T    Then       c    1    +      c    2    +      c    3    =  2        c    2    +      c    3    =  5        c    3    =  1    ∴      c    2    =  4   and       c    1    =  −  3    ∴  (  2  ,  5  ,  1      )    T    =  −  3  (  1  ,  0  ,  0      )    T    +  4  (  1  ,  1  ,  0      )    T    +  (  1  ,  1  ,  1      )    T  The coordinated of   w  =  (  2  ,  5  ,  1      )    T   with respect to the B is  (  −  3  ,  4  ,  1      )    T

Find the coordinates of w=(2, 5, 1)^T with respect to the basis B = {(1,0,0)^T, (1, 1,0)^T, (1,1,1)^T}. Enter coordinates into the answer box below, separated with commas.

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