Signal and System as Vectors
To solve forward and inverse problems, we need mathematical tools to formulate them. Since it is convenient to use vectors to represent multiple system parameters, variables, inputs (or excitations) and outputs (or measurements), we first study vector spaces. Then, we introduce vector calculus to express interrelations among them based on the underlying physical principles. To solve a nonlinear inverse problem, we often linearize an associated forward problem to utilize the numerous mathematical tools of the linear system. After introducing such an approximation method, we review mathematical techniques to deal with a linear system of equations and linear transformations.
We denote a signal, variable or system parameter as f(r, t), which is a function of position r = (x, y, z) and time t. To deal with a number n of signals, we adopt the vector notation (f1(r, t), …, fn(r, t)). We may also set vectors as (f(r1, t), …, f(rn, t)), (f(r, t1), …, f(r, tn)) and so on. We consider a set of all possible such vectors as a subset of a vector space. In the vector space framework, we can add and subtract vectors and multiply vectors by numbers. Establishing the concept of a subspace, we can project a vector into a subspace to extract core information or to eliminate unnecessary information. To analyze a vector, we may decompose it as a linear combination of basic elements, which we handle as a basis or coordinate of a subspace.