In case of, for example, the discrete fourier transform or the discrete cosine transform, it is automatic to build the basis of the transform once the dimension is selected.
So, if I want to compute one of these transforms on a generic sequence of length $n$, I know exactly how to build the $n\times n$ matrix that computes the transformation, since the basis functions are given in analytic form.
But, let suppose that I have a basis of a vector space of dimension $n$. If I ask you which is the basis of the vector space of dimension $n+m$ "nearer" to the previous basis, how do you answer? Has this question a meaning?
Another related question can be: if I have a vector in a vector space of dimension $n$, can I compute a new vector in a new vector space of dimension $n+m$ such that the two vectors are more related possible each other.
I know I am not precise to define concepts as "nearer" and "more related", but it is exactly the point of my question.
The answer to your question is that the New vector in that n+m vector space is the same vector of your n-dimensional vector space due to the fact that as your new vector space is constructed from the original one, so your original vector is contained there too.
Consider the folllwing example: imagine that your initial vector is (1,1,1) the $\mathbb{R}^3$ space, the if you add another dimension to your vector space and form $\mathbb{R}^4$, then vector (1,1,1,0) is contained in the new space and is the same vector as the original one. This happens because the original space of dimension n is a subspace of the n+m space, making that all vectors that are contained there are also contained in the new more dimensional space.
The concept of nearer vector would have sense in the case that you want to get a vector in a n-m vector space, because the original vector might nit be included in that n-m subspace, then being the nearest vector $\sum_{i=1}^{n-m}<x_n,e_i>e_i$ where $e_i$ are the vector basis of the n-m new space.
You higher dimensional problem would make sense if you put restrictions to the new vector space in the new dimensions, and the solution to it would depend om those restrictions.