Start Maple from Start Menu/Program Files or using the icon
in f:\math115, or double click on 115lab5base.mws

Load the linear algebra progams with the command:
>    with(linalg):

First, let us quickly solve the questions of the midterm 
using these matrices. The questions are in f:\math115\11504mid.pdf
T:=matrix(3,3,[-13,10,15,-30,27,45,10,-10,-18]);
R:=matrix(4,4,[2,x,1,0,-1,1,0,1,2,2,x-1,0,4*x-1,1,0,-1]);
Q:=matrix(9,9,1):Q:=mulrow(Q,9,-8);
S:=matrix(2,2,[-61/2,42,-45/2,31]);

Here is something from the book which is too tedious to do without a computer:

First, generate a random 4x4 matrix M (make sure it has rank 4)
and 4x1 vector b and solve the equation M x = b.

Then form four new matrices, each time replacing different columns
of M with b. You can use the commands delcols, augment and col here.
Calculate the determinants of these 4 matrices and
divide each by the determinant of M. Do you notice any connection
with the vector x found before? This is Cramer's Rule.

Let us now look at vectors, we can define them in a shorthand way:
> v1:=vector([1,2,3,-1]);
> v2:=vector([-1,2,2,-1]);

We can use the matadd commands to add vectors, or just use
> evalm(v1-v2);
Note that if we don't use evalm it displays them symbolically.
Verify that 1/2 times 5v1-4v2 is equal to 5/2 v1 - 2 v2 as expected.

We can take the dot product of vectors:
> dotprod(v1,v2);
Project v2 onto v1 to find an e2 which is orthogonal to v1
> e2:=v2-dotprod(v2,v1)/dotprod(v1,v1)*v1;

Note that we use dotprod(v1,v1) for the length of v1 squared.

If we define 
> v3:=vector([-1,0,-3,2]);
show that it is obtuse with both of the other vectors.

Using the Gram-Schmidt method, find e3 from v3, v2 and v1.

Find a fourth orthogonal vector, starting with v4:=vector([0,0,1,0]);
then again with
> v4:=vector([-2,-1,2,3]);
Do the same with v4:=vector([1,0,0,0]); and then try a random v4.
Try once more with v4:=vector([7, 12, 31, -16]);
What goes wrong here?