Math 22 Fall 2004

Linear Algebra with Applications

Vector and Matrix Equations

September 29, 2004

    Load the package for doing Linear Algebra

>	with(Student[LinearAlgebra]):

Warning, the protected name . has been redefined and unprotected

Example 1: Solve a vector equation

x1*a1 + x2*a2 = b

>	a1 := <-1, 3, -2>: a2 := <3, 2, -3>: b := <5, -4, 1>: a1, a2, b;

     Define an (augmented) matrix of the equation

>	A := <a1 | a2 | b>;

    Reduce it to an echelon form

>	A := AddRow(A, 2, 1, 3): A := AddRow(A, 3, 1, -2);

>	A := MultiplyRow(A, 2, 1/11): A := AddRow(A, 3, 2, 9);

    ... and to the reduced echelon form

>	A := AddRow(A, 1, 2, -3);

    Solve the system now

>	x := LinearSolve(A);

    Test our solution

>	x[1] * a1 + x[2] * a2 = b;

Example 2: Solve a matrix equation Ax=b

>	A := <<1, 3, -2> | <2, -1, 3> | <-2, -3, 1>>: b := Vector(3, symbol = v): M := <A | b>: A, b, M;

    Solve this system

>	M := AddRow(M, 2, 1, -3): M := AddRow(M, 3, 1, 2);

>	M := AddRow(M, 3, 2, 1);

   Conclusion: this system is not consistent for every b

    It is easy to find v1, v2, v3 such that v3-v1+v2 is nonzero

>