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

Sattelhofsk

Sattelhofsk

Answered question

2022-06-14

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 ?

Answer & Explanation

Korotnokby

Korotnokby

Beginner2022-06-15Added 19 answers

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

veirarer

Beginner2022-06-16Added 9 answers

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 B
Under 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.

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