Define what it means for a set \\ to be a vector space over F, giving all the operations on \\ and the properties that these operations must satisfy. Then, state the Subspace Test used to determine whether a subset S of V is a subspace. Define what is means for a function T : V - W between vector spaces to be a linear transformation. Define what it means for a subset B of V to be an ordered basis for V. Then, if B is an ordered basis for V and C is an ordered basis for W (and both spaces are finite-dimensional), define the entries of the matrix representation c[T]s- Given a vector space V, define what it means for a function (., .) : V x V - V to be an inner product on V. Then, define what it means for a subset S of V to be an orthogonal set with respect to this inner product. Finally, define what it means for a linear transformation T : V - V to preserve inner products