Hi Alan,
Mathematica has a "front end" and a "kernel." The front end is the user interface; The kernel performs the computations. The values of variables, the real state of the system, is in the kernel. Notebooks are creatures of the front end. There can be several open. You can close them and open new ones. But there is only one kernel, and all notebooks use it for computation. If you have several notebooks open, and one notebook defines x=3, that is in the kernel. Any other notebook wanting to use x will find it is equal to 3. If you close the notebooks, but not the kernel -- that is you do not quit Mathematica -- a newly opened notebook will still find all the old definitions in existence.
To start fresh you can quit Mathematica entirely and restart it. Or you can go to the menu bar and select Evaluation->Quit Kernel. You can also Clear all variables in the Global context with Clear["Global`*] as I do below.
I highly recommend sometime with The Virtual Book
Also the help system in Mathematica is excellent. For some specific answers see the code below:
In[1]:= Clear["Global`*"]
In[2]:= (* a general matrix is just a regular list of lists *)
m = {{a, b}, {c, d}};
In[3]:= (* The dot operator is used for multiplication of vectors, \
matrices, and tensors.*)
m.{x, y}
Out[3]= {a x + b y, c x + d y}
In[4]:= (* You just create them at need. *)
{{1, 2}, {3, 4}}.{5, 6}
Out[4]= {17, 39}
In[5]:= {1, 2}.{3, 4}
Out[5]= 11