# Change of Basis for Bilinear map/form

1. Dec 21, 2011

### The1337gamer

I was wondering if some could check these notes which i had written, i seem to think there is a mistake in the Bilinear map part below, which i have pointed out in red. Just need someone to clarify whether i am correct about this mistake or whether my understanding is mistaken as it confused me when i read over it. Also 'wrt' is shorthand for 'with respect to'.

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Notes for change of basis of a linear map:

V is a vector space, dimension n. We have 2 bases:
e1,e2,...,en and
e1',e2',...,en'

Then if we want the change of basis matrix from
old basis : e1,e2,...,en to
new basis: e1',e2',...,en'

we write the old basis in term of the new basis:

e1 = a11e1'+.....+an1en'
.
.
.
en= a1ne1'+.....+annen'

Then P = (aij) 1<= i,j <= n is the change of basis matrix from old basis to new basis.

So if we have a vector v wrt old basis and v' wrt new basis then Pv = v'

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For Bilinear Form my notes say:

Let V (dim n), W (dim n) be vector spaces.

e1,e2,...,en = old basis for V
e1',e2',...,en' = new basis for V

f1,f2,...,fm = old basis for W
f1',f2',...,fm' = new basis for W

Let P be the change of basis matrix from old basis {ei} to new basis {ei'}, let Q be the change of basis matrix from old basis {fi} to {fi'}.

Then v, a vector in V has coordinates v wrt old basis and v' wrt new basis.
And w, a vector in W has coordinates w wrt old basis and w' wrt to new basis.

So v = Pv' and w=Qw' (This is where i think the mistake is as it contradicts the part above, i think it should be Pv = v')

Let the matrix A be T in old basis and B be T in the new basis, then:

T(v,w) = vTAw = (v')TB(w')

So

T(v,w) = vTAw = (Pv')TA(Qw') = (v')TPTAQ(w') = (v')TB(w')

So PTAQ = B