Hi, am I right in thinking that for a L.T. T: U → V represented by Au = v, u, v refers to the coordinates of a vector within vector spaces U, V with respect to that space’s basis, represented as a column vector? I’m getting a little confused since T(u) = v refers to u, v as vectors and not their coordinates, but then Au = v wouldn’t make sense in a polynomial space where, for example, u = α x²+ βx+ γ, unless u, v refers to the coordinates in column form.
If so, does this mean that given a vector space V over a field K, we are essentially using elements of K as entries in a coordinate system with respect to the basis of V s.t. each distinct element in V has 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) in K³ and the polynomial α x²+ βx+ γ in K[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 one coordinate system to another while applying a given L.T. on the way.
Thanks very much.