I need some help or at least some assurance that my thinking on linear transformations and their matrix representations is correct.

I assume when we specify a linear transformation eg F(x,y, z) = (3x +y, y+z, 2x-3z) for example, that this is specified by its action on the variables and is not with respect to any basis.

However when we specify the matrix of a linear transformation T: V --> W that this is with respect to a basis in V and a basis in W

Of course if we have a linear transformation S: V -->V it could be that the two bases are the same.

If we are dealing in Euclidean space and no basis is mentioned regarding the matrix of a linear transformation, then I am assuming the standard bases are assumed.

Can someone either confirm I am correct in my thinking or point out the errors in my thinking?

Peter