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'.(adsbygoogle = window.adsbygoogle || []).push({});

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Notes for change of basis of a linear map:

V is a vector space, dimension n. We have 2 bases:

e_{1},e_{2},...,e_{n}and

e_{1}',e_{2}',...,e_{n}'

Then if we want the change of basis matrix from

old basis : e_{1},e_{2},...,e_{n}to

new basis: e_{1}',e_{2}',...,e_{n}'

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

e_{1}= a_{11}e_{1}'+.....+a_{n}1e_{n}'

.

.

.

e_{n}= a_{1n}e1'+.....+a_{nn}e_{n}'

Then P = (a_{ij}) 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.

e_{1},e_{2},...,e_{n}= old basis for V

e_{1}',e_{2}',...,e_{n}' = new basis for V

f_{1},f_{2},...,f_{m}= old basis for W

f_{1}',f_{2}',...,f_{m}' = new basis for W

Let P be the change of basis matrix from old basis {e_{i}} to new basis {e_{i}'}, let Q be the change of basis matrix from old basis {f_{i}} to {f_{i}'}.

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) = v^{T}Aw = (v')^{T}B(w')

So

T(v,w) = v^{T}Aw = (Pv')^{T}A(Qw') = (v')^{T}P^{T}AQ(w') = (v')^{T}B(w')

So P^{T}AQ = B

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# Change of Basis for Bilinear map/form

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