If I have vec(w)_1=(2,3) and vec(w)_2=(1,1), but they are relative to the basis vec(u) =(1,1), vec(v) =(1,−1). How do I find the scalar product of w_1 and w_2?

gheadarce

gheadarce

Answered question

2022-11-24

If I have w 1 = ( 2 , 3 ) and w 2 = ( 1 , 1 ), but they are relative to the basis u = ( 1 , 1 ) , v = ( 1 , 1 ). How do I find the scalar product of w 1 and w 2 ?
I know that w 1 , w 2 = 2 1 + 3 1 when the basis are orthogonal, but that is not the case here. Would I say that w 1 = 2 1 + 3 1 and w 2 = 1 1 + 1 1? If so, then how do I proceed from here?

Answer & Explanation

Dakota Murillo

Dakota Murillo

Beginner2022-11-25Added 6 answers

We have that in the standard basis
w 1 , w 2 = w 1 T w 2
and indicating with M the matrix for the change of basis from the the new basis B to the standard basis we have
w 1 , w 2 = ( M w 1 , B ) T M w 2 , B = w 1 , B T M T M w 2 , B
therefore the key point is to find the matrix M.

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