I am having trouble understanding the concept of AI.
Is AI just a software that uses Probability, Linear Algebra and some advanced Mathematics to make independent decision ?
Please give me some simple examples of AI.
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.
An easy example is Artificial Intelligence in (video) games ("Game AI").
In 1997 Deep Blue won a six games match against Garry Kasparov and (in my opionion) that match is a milestone in the A.I. history: for the first time a computer beated a human in one of the most ancient, famous and "intelligent" game (though recently the relation between chess skill and IQ has been disproved). Nowadays most people think that a computer could easily beat the best chess players.
Modern video games often make large use of "A.I." and you can find a lot of documentation/references online about the techniques and approaches used.
There is no proof that a computer can make an "independent decision". In fact, there is no real proof that humans can, either...
Perhaps the AI "holy grail" is the Turing-Test.
In practice, AI covers, or touches a wide range of topics, from theorem-proving, planning, decision-making, multi-agent systems, some machine learning, and many more. Try reading this.
As for your description of AI: not every AI program uses any of the concepts you suggested, and not every program that uses them is considered AI. So in that sense - no. But any program that is implemented to solve some task which is considered as AI (e.g. an autonomous car driver), is considered as AI.
Also, this rather amorphic concept is often referred to as "weak AI", as opposed to "strong AI", which refers to machines that can "think for themselves", which is something that we don't have (yet).