Lecture 6 Notes

# V 1 v 2 v 3 is a linearly dependent set 2 10 0 x1 01

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: t set 2 10 0 x1  01 18 0 Í x2  00 (b) Reduced echelon form: 33 0 0 x3 Let x 3  _____ (any nonzero number). Then x 1  _____ and x 2  _____. 1 ____ 3 5 2  ____ 5 3  ____ 9 9 0  3 0 0 or ____v 1  ____v 2  ____v 3  0 (one possible linear dependence relation) 3 Linear Independence of Matrix Columns A linear dependence relation such as 1 33 3 2  18 5 5 3 1 9 9 0  0 3 0 can be written as the matrix equation: 12 3 33 35 9 18 59 3 1 0  0 . 0 Each linear dependence relation among the columns of A corresponds to a nontrivial solution to Ax  0. The columns of matrix A are linearly independent if and only if the equation Ax  0 has only the trivial solution. 4 Special Cases Sometimes we can determine linear independence of a set with minimal effort. 1. A Set of One Vector Consider the set containing one nonzero vector: v 1 The only solution to x 1 v 1  0 is x 1  _____. So v 1 is linearly independent when v 1 0. 5 2. A Set of Two Vectors EXAMPLE u1  Let 2 1 , u2  4 2 , v1  2 1 , v2...
View Full Document

## This document was uploaded on 03/03/2014 for the course MTH 215 at Rhode Island.

Ask a homework question - tutors are online