BASIS AND DIMENSION OF A VECTOR SPACE

Linear Algebra Project

Objective:

To introduce the concepts of basis and dimension of a vector space. Maple will carry out the details of the Gauss-Jordan method.

Solved example:

Let A = . The solution set for A x = 0 forms a vector space. Find a basis of this vector space. What is the dimension of this vector space?

Solution:

We will solve the system using the Gauss-Jordan method.

> with(linalg):

> A:=matrix([[-1, -1, -2, 3, 1], [-9, 5, -4, -1, -5], [7, -5, 2, 3, 5]]);

> gaussjord(A);=

We see that the only leading variables are and . Taking as , as , and as , we can write the first two rows as equations for and .

The solution is x = ( , , , , ), where = - +

= - +2 +

.

Any solution vector x can therefore be represented as

x = = u + v + w ,

which means that the solution space is spanned by { u , v , w }. To see whether or not the vectors form a basis for the solution space we need to find out whether

u , v , w are linearly independent.

The vectors u , v , and w are linearly independent if and only if the equation u + v + w = 0 has the trivial solution only.

> B:=matrix(5,3,[0,1,-1,1,2,-1,0,0,1,0,1,0,1,0,0]);

> gaussjord(B);

The solution is = = = 0. The vectors u , v , and w are linearly independent.

Vectors { u , v , w } form a base. The dimension of the space is three.

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Assignment

Use Maple to solve systems of equations.

Problem 1.

Find a basis for S = span{ r, u, v, w }, where r = (1, 3, 2, -5), u = (0, 1, 5, -3), v = (4, 1, 1, -1), and w = (-2, 5, 3, -9). What is the dimension of S ?

Problem 2.

Find a basis for the solution set of 2x - y + 4z = 0. What is the dimension of the solution space?

Problem 3.

Consider the solution set of the system of equations 2 x - y + 4 z = 0

x + 3 z = 0 .

a) Show that this set forms a subspace of .

b) Find the base for the solution space.

c) What is the dimension of this vector space?

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MSIP Grant #P120A80089-98: "Three Urban Calculus Reform Programs: Adopting the Best," 1998-2001; MSEIP Grant #P120AA010031: "Four Colleges: Calculus + Enhancements", 2001-2004