Hi, am I right in thinking that for a L.T.T:U→Vrepresented byAu=v,u,vrefers to thecoordinatesof a vector within vector spacesU, Vwith respect to that space’s basis, represented as a column vector? I’m getting a little confused sinceT(u) =vrefers tou,vas vectors and not their coordinates, but thenAu=vwouldn’t make sense in a polynomial space where, for example,u=αx²+βx+γ, unlessu,vrefers to thecoordinatesin column form.

If so, does this mean that given a vector spaceVover a fieldK,we are essentially using elements ofKas entries in a coordinate system with respect to the basis ofVs.t. each distinct element inVhas a unique coordinate representation or equivalently, is a unique L.C. of basis vectors?

So does this mean that all spaces over the same field with the same dimension are isomorphic? Since, for example, the row vector (α,β,γ) inK³ and the polynomialαx²+βx+γinK[x]≤2 both have the same coordinates (namely (α,β,γ)) and so a L.T. to coordinates (α’,β’,γ’) would be represented by the same matrix for both spaces since the matrix only transforms onecoordinatesystem to another while applying a given L.T. on the way.

Thanks very much.