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sebadillab0 2022-07-07 Answered
Lets say, there is a transformation: T : n m transforming a vector in V to W. Now the transformation matrix,
A = [ a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . a n 1 a n 2 . . . a n 3 ]
The basis vectors of V are v 1 , v 2 , v 3 . . . , v n which are all non standard vectors and similarly w 1 , w 2 , . . . , w m
My question is, in the absence of the basis vectors being standard vectors what is the procedure of finding T
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Answers (1)

Elianna Wilkinson
Answered 2022-07-08 Author has 11 answers
use the example in your comment: M = ( 4 2 1 0 1 3 ) , with respect to the bases
β = { v 1 = [ 1 , 0 , 0 ] , v 2 = [ 1 , 1 , 0 ] , v 3 = [ 1 , 1 , 1 ] }
and
β = { w 1 = [ 1 , 0 ] , w 2 = [ 1 , 1 ] } .
This transformation takes as input a column vector
v = ( α 1 α 2 α 3 )
representing the linear combination α 1 v 1 + α 2 v 2 + α 3 v 3 of basis vectors in β and produces as output the product
M v = ( 4 2 1 0 1 3 ) ( α 1 α 2 α 3 ) = ( 4 α 1 + 2 α 2 + α 3 α 2 + 3 α 3 ) .
The entries 4 α 1 + 2 α 2 + α 3 and α 2 + 3 α 3 are then to be interpreted as coefficients of the vectors w 1 and w 2 in β :
T ( v ) = ( 4 α 1 + 2 α 2 + α 3 ) w 1 + ( α 2 + 3 α 3 ) w 2 .
If you want to know what this looks like in terms of the standard basis { [ 1 , 0 ] , [ 0 , 1 ] }, just multiply out:
Note that this is exactly what you get from the product
A M v = ( 1 1 0 1 ) ( 4 2 1 0 1 3 ) ( α 1 α 2 α 3 ) ,
where A = ( 1 1 0 1 ) is a change-of-basis matrix: it translates a representation in terms of β into one in terms of the standard basis. It’s easy to construct this change-of-basis matrix: its columns are just the representations of w 1 and w 2 in terms of the standard basis.
It follows that if you start with v, representing a vector in R 3 in terms of the basis β, and multiply it by the matrix
A M = ( 4 3 4 0 1 3 ) ,
you get T ( v ) expressed in terms of the standard basis for R 2 . Perhaps, though, you want to be able to input v in terms of the standard basis for R 3 . Then you need another change-of-basis matrix, this time to convert from the standard basis for R 3 to the basis β We already know how to go the other way: to transform from a representation in terms of β to one standard coordinates, multiply by the matrix
B = ( 1 1 1 0 1 1 0 0 1 )
whose columns are the representations of w 1 , w 2 and w 3 in terms of the standard basis. (In other words, do exactly what we did to get A.) If you take the vector α 1 v 1 + α 2 v 2 + α 3 v 3 represented by the matrix
v = ( α 1 α 2 α 3 )
in terms of the basis β , you can find its representation in terms of the standard basis by multiplying by B to get
B v = ( 1 1 1 0 1 1 0 0 1 ) ( α 1 α 2 α 3 ) = ( α 1 + α 2 + α 3 α 2 + α 3 α 3 ) .
Unfortunately, this isn’t quite what we need: we want to start with a vector v in standard coordinates and convert it to one in β coordinates so that we can multiply by A M and get T ( v ) in standard coordinates. That requires changing base from standard to β; multiplying by B goes in the opposite direction, from β coordinates to standard ones. As you might expect, the matrix that does the change of basis in the other direction is B 1 , which I’ll let you compute for yourself. Once you have it, you can express T in terms of a matrix multiplication that involves standard coordinates on both ends:
T ( v ) = A M B 1 v
gives T ( v ) in standard R 2 coordinates if v is expressed in standard R 3 coordinates.

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