Simple examples of Artificial Intelligence
Artificial Intelligence is a catch-all term for a large collection of everything from philosophical questions down to very specific programming techniques.

At heart, AI comes from the question is it possible to build a machine that displays intelligent behavior to the same level as human beings? This is often clarified with examples like, can we build a machine that plays chess, composes a sonnet, tells a joke or recognizes kittens. Investigating this question has led to the distinctions between strong and weak AI, and thought-experiments like the Turing test and the Chinese room.

On the more practical side, there is a collection of techniques that are being used to (partially) solve some of these problems. For instance, we can build a machine that beats any human at chess or checkers. Computers can recognize objects in images with reasonable accuracy. They can diagnose diseases, sometimes better than doctors, they can parse a sentence into its constituents and make a reasonable mapping to the meaning behind it.

Many of these algorithms and techniques use probability, linear algebra, logic and a variety of other subjects, but that's just a matter of using what works. There are plenty of systems that use linear algebra that have nothing to do with AI. What makes a technique AI is that, in some sense, it attempts to behave intelligently. 

A precise definition beyond that is difficult to give. The trouble with defining AI is that it's the study of something that doesn't exist yet, and we don't know that it ever will. But we're pretty sure that we'll know it if we see it.