Change of Basis of Linear Transformation

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• Nov 2nd 2012, 08:21 PM
sfspitfire23
Change of Basis of Linear Transformation
Hi all,

Let B be a basis for R^2. Let L be the linear transformation represented in standard coordinates by the matrix R. Then, to represent L with respect to the standard coordinates, we take A^(-1)RA where A is a column matrix of the B vectors.

Now, what if R was expressed in terms of the basis B and we want to express L with respect to the standard coordinates? How would we go this way?

Thanks
• Nov 2nd 2012, 10:52 PM
chiro
Re: Change of Basis of Linear Transformation
Hey sfspitfire23.

The easiest way to look at this is to go from first principles.

Now in one basis we have some vector v in Basis B, and in another basis we have v' in Basis B'. Let x be the true vector that is invariant to both bases: then this implies

Bx = v and
B'x = v'. If B and B' are both basis then B and B' are invertible giving

x = B^(-1)v = B'^(-1)v'. Writing these equations to get v in terms of v' (and vice versa) gives:

B*B'^(-1)v' = v and
B'*B^(-1)v = v'

Now a linear transformation acting on R^2 is simply Ax = b taking x and mapping it to b, and the above can relate Av to Av' in a straight-forward manner.
• Nov 3rd 2012, 05:41 AM
Deveno
Re: Change of Basis of Linear Transformation
suppose B and B' are two bases for a vector space V, and L:V-->V is a linear transformation.

if [L]B = R, and the change-of-basis matrix from B' to B is A (whose columns consist of the basis vectors of B' expressed in B-coordinates), then:

[L]B' = A-1RA, as you correctly deduced.

it therefore stands to reason that:

R = A[L]B'A-1 (use matrix multiplication).

here is how it works:

A-1([v]B) = [v]B' (since A changes B'-vectors to B-vectors, A-1 "changes them back" to B'-vectors).

[L]B'([v]B') = [Lv]B' (the matrix for L in the basis B' takes v in B'-coordinates to Lv in B'-coordinates)

A([Lv]B') = [Lv]B (A turns Lv in B'-coordinates to Lv in B-coordinates).

so A[L]B'A-1([v]B') = A([L]B'(A-1([v]B))) = A([L]B'([v]B')) = A([Lv]B') = [Lv]B, that is:

A[L]B'A-1 is a linear transformation which takes [v]B to [Lv]B for every v in V, and so is the matrix for L in the basis B, that is: R.

the algebra "looks prettier" if we call the matrix for L in the basis B', S.

then S = A-1RA so:

AS = AA-1RA = IRA = RA

ASA-1 = RAA-1 = RI = R