Start Maple 2016 from the desktop, Start Menu/All apps/M or 
using the icon in f:\"math 2301". 

Maple has two modes, Worksheet and Document. You can use either,
but Document is more like a calculator rather than a word 
processor. To start one or the other use File/New

Every Maple command should end with a semi-colon ";" for instance
>   2+1;
The answer will appear in the middle of the line following.
If you forget Maple wont actually complain, usually, though.

Every time I give you a command to use I will give the > symbol in front.
You are free to copy the command from this file and paste into Maple,
this will ensure you don't make a mistake when typing it in.

Multiplication of numbers uses *, division uses / and powers are ^.
Pick several integers and combine them using these two symbols and brackets
and make sure you get the answers you expect.

Don't forget the semi colon! Notice that Maple does not use 
decimals by default, and neither will I, in class.

We can use ":=" to assign a value to a name, usually a letter
>   a := -5;
This is case sensitive, so we can set 
>   A := 4;

A common mistake is to forget the : since Maple will still show "A=4"
but A will not be assigned the value 4 in Maple's memory.

Verify that A-a is not 0 and a/A gives a fraction, not a decimal.

For an unknown variable, Maple will just report that name:
>   a+b;

Now set b to be some randomly chosen integer and note that
the line above doesn't change but when you again ask for 
>   a+b;
it should give a number now instead. Maple works sequentially so
I encourage you to work down the worksheet without 
editing earlier lines so that the confusion is lessened. 

(
If you want to you can go back up the worksheet and insert 
new lines if necessary using control-j or control-k.
You can reset a variable's value to unknown using 
> a:='a';
)

 In the other labs I will use capital letters for matrices, sometimes 
with a number after them, and lower case letters for numbers, but 
it isn't mandatory.


When I introduce a Maple function it is a good idea to read the help page
for that function. You can do that for the function sqrt by 
visiting 
  http://www.maplesoft.com/support/help/Maple/view.aspx?path=sqrt
and can search for any other function in that webpage too.

You can also do it by using the Maple "Help" menu or by typing
> ?sqrt
However, ? can be slow to work, and might crash your computer.
To recover, press the octagonal red button marked "stop".

To exit from the help page, close the subwindow, use alt-tab or the 
Window menu to move between them.

Use sqrt to determine the square roots of A and 10.
Notice that, similarly to fractions, square roots are left as is too, 
unless the number is a perfect square, as this is most accurate.


We are now ready to do some matrix algebra...
Load all of the linear algebra progams with the command:
>    with(LinearAlgebra);

All the new commands listed are now known by Maple, we will use some.

We can create a 3x4 augmented matrix by using the following command
>    B:=<<3,-6,8>|<-1,-6,3>|<2,-5,6>|<7,3,5>>;
It gives the matrix in terms of its columns. Even though we are using this
to represent a system of equations Maple doesn't give a dotted line,
but we should try to remember it is there.

Notice that we again use ":=" to assign a value to B. I recommend using
a capital letter for all matrices, but Maple will not insist on that. 
Recall that Maple is case sensitive, so B is different from b.

Note that all of the LinearAlgebra commands also begin with a capital 
letter, although "matrix" is also defined, it will not always work properly.

We can do the three different row operations as follows:
> B1:=RowOperation(B,[1,3]);
will swap row 1 and row 3 of B and make it into a new matrix B1.
Recall that this operation is not needed for our method.
> B2:=RowOperation(B1,2,-1);
would take this new B1 and multiply row 2 by -1 and make B2
This operation is only occasionally needed for our method.

The following is the main row operation we will be using:
> B3:=RowOperation(B2,[2,3],4);
will replace row 2 of B2 by itself plus 4 times row 3 to make B3
(
In general N:=RowOperation(M,[p,q],r) does takes matrix M and
turns it into matrix N using "Row p <-- Row p + r times Row q"
)
  
I recommend using this scheme of naming each matrix sequentially, and
try to not reuse any name, if you don't get the matrix you wanted, just
use the next Bnumber matrix with the previous one.

Note that the B3 operation will not give you the desired 0 in row 2
of column 2, so change the multiplier used to make a B4 with a 0 from B3, 
then find a new row operation for B4 which will make a 0 in column 2 of B5.

Note that we should now have one column with only one 1 and all other 
elements 0. However, the other two columns do not have a 1 in. Find a 
series of row operations using only row 1 and row 2 which eventually create 
a new 1 in the first or third column but which does not introduce fractions. 

Use the 1 created to create two more 0s it is column and hence find the 
solution to the system of equations represented by B, and check your answer
using paper and pen.

Note that you can also use the "multiplication of a row" operation to 
create a 1 immediately, but at the expense of creating fractions. 
Maple can cope with fractions ok, but I find that it means that more 
errors will occur if you try them by hand...


Define
> J:=Matrix([[-3, 1, -4], [2, 4, 3], [2, -1, 0]]);
and
> w:=Matrix(3,1,[1,8,4]);

We can augment a column to our matrix J (but not make a dotted 
line unfortunately) using the "|" symbol which is shift "\":
> J0:=< J | w >;

Use the command Pivot (use help to find out how it works)
to produce two zeros in column 2, then continue using Pivot 
until you have six zeros on the left of the imaginary dotted line.

Try to see what the answer to this system of equation was as
> v:= < number, number, number >;
and check that you get w with the command "Multiply" as follows:

> Multiply(J,v);

If it isn't right, try predicting again until it works.
This is matrix multiplication which we will be meeting soon!

Now set 
> K:=Matrix([[2, 1, 3], [3, 2, 4], [1, -3, 5]]);
and 
> u:=<1,1,4>;
> K0:=< K | u >;

Pivot and use Row Operations to show that there are infinite solutions.
Identify two different solutions and check that they are correct, and
then identify the particular and homogeneous solutions and verify that
they have the property mentioned in class.


If you have time you can create a random augmented matrix as follows:

>  Seed:=randomize(): C:=RandomMatrix(3,4,generator=rand(-6..5));
(
It will be a 3x3 matrix alongside a 3x1 matrix of numbers, all between -6 and 5.
)
and then create C1, C2, etc. using pivots or row operations 
until you have as many zeros as possible. Note that most likely your final 
solution will have fractions in it though.


Once you are done, call me over and I'll make sure you've 
got all of your answers correct. You can save your work in your H:
drive and it will be available for you to consult in future labs.