Solving Linear Algebra problems using Maple. 

>

restart;

 

Start by loading the linear algebra package 

>	with(LinearAlgebra);

 

Vectors and matrices are defined as follows. 

>	x:=Vector([1,2,3]);

 

>	A:=Matrix([[1,2,1],[2,4,-2]]);

 

Matrix-vector product 

>

b:=A.x;

 

Finding the echelon form (not unique) 

>	GaussianElimination(A);

 

Finding the reduced row echelon form (unique) 

>	ReducedRowEchelonForm(A);

 

Warning, does not work for matrices with parameters. Two examples: 

>	B:=Matrix([[1,2,1],[2,4,h]]);

 

>	ReducedRowEchelonForm(B);

 

This is wrong for h=2. 

>	C:=Matrix([[u,v],[w,z]]);

 

>	ReducedRowEchelonForm(C);

 

This is wrong for many different values of the four parameters in this matrix. 

Solving a system of linear equations 

Define the coefficient matrix and the right hand side, as in the following examples. Then use the linear solver. 

>	F:=Matrix([[1,2,3],[3,1,2],[-1,-1,-1]]);

 

>	c:=Vector([1,2,3]);

 

>	LinearSolve(F,c);

 

If the solution is not unique, then Maple returns the solution in parametrized form. 

>	G:=Matrix([[1,2,3],[2,1,4]]); d:=Vector([2,2]);

 

>	LinearSolve(G,d);

 

Maple uses a particular notation for variables it defines. They always start with an underscore _ 

You can make Maple use any name you choose, giving it as a string in the input. The above example again, with this option. 

>	LinearSolve(G,d,free='t');

 

>