General concepts regarding linear transformations and matrices

Hi, am I right in thinking that for a L.T. *T*: *U **→** V* represented by *A***u **= **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 *A***u **= **v **wouldn’t make sense in a polynomial space where, for example, **u** = *α **x*²+ *β**x*+* 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 *K*³ and the polynomial *α **x*²+ *β**x*+* 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.