Vectors, Functions, and Diagonal Matrices

Later on in this book, we are going to encounter vector spaces, in which the vectors are functions of continuous variables. A lot of students get confused transferring what they know about linear algebra into the context of function spaces, so it is worth explaining that we can always think of linear algebra in terms of functions. If v_n are the components of a vector |v〉 in some basis, then we can associate with them a complex valued function, defined on the finite set of integers from 1 to N, by the rule

The space of all such functions is a vector space, since we can add them and multiply them
by complex numbers, and the scalar product takes the form

The basis vectors are functions which vanish on all integers between 1 and N except for one. Linear operations become operations on functions. A particularly interesting one is the finite difference operator

If we take a limit where we think of N going to infinity and define x = n/N , then x becomes a continuous variable in the limit and N
approaches the derivative operator, when it acts on differentiable functions of x. You should write out the matrix corresponding to the difference operator
to make sure you understand this. Note that this correspondence between vectors, and functions defined on the discrete set 1 . . .N, depended on a choice of basis. Different choices of basis will give different functions for the same vector |vi. When we get to continuous variables, we will see that the famous Fourier Transform is just a relation between the functions that represent a vector in two different bases.
Another confusion that arises when thinking about functions as elements of the space is that functions also act as multiplication operators on the space of functions. This also has a finite dimensional analog. Given a vector v_n, or the associated function f_v(n) one can construct a diagonal matrix, whose n-th diagonal element is v_n. When one acts with that matrix on a vector whose components are wn, then one gets the vector whose components are v_nw_n = f_v(n)w_n. The function corresponding to this new vector is the product f_v(n)f_w(n).
These remarks may seem sort of silly in the finite dimensional context, but they are the bridge that allows you to make the transition to thinking about function spaces.