Linear systems

In this course we will focus much of our attention on linear systems. When our input and output signals are vectors, then the system is a linear operator .

Suppose that $L:X\to Y$ is a linear operator from a vector space $X$ to a vector space $Y$ . If $X$ and $Y$ are normed vector spaces, then we can also define a norm on $L$ . Specifically, we can let

An operator for which ${∥L∥}_{\mathcal{L}(X,Y)}<\infty$ is called a bounded operator .

One can show that ${∥·∥}_{\mathcal{L}(X,Y)}$ satisfies the requirements of a valid norm. In fact $\mathcal{L}(X,Y)=\left\{\text{bounded linear operators from}X\mathrm{to}Y\right\}$ is itself a normed vector space! If $Y$ is a Banach space, then so is $\mathcal{L}(X,Y)$ !

Bounded linear operators are common in DSP—they are “safe” in that “normal” inputs are guaranteed to not make your system explode.

Are there any common systems that are unbounded? Not in finite dimensions, but in infinite dimensions there are plenty of examples!

A very important class of linear operators are those for which $X=Y$ . In this case we have the following important definition.

Definition 1

Eigenvalues and eigenvectors tell you a lot about a system (more on this later!). While they can sometimes be tricky to calculate (unless you knowthe eig command in Matlab), we will see that as engineers we can usually get away with the time-honored method of “guess and check”.