# Hence x now setting with solution given by x 17 the

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: dent. 10. Write the given vectors as columns of the matrix X The four vectors are necessarily linearly dependent. Hence there are nonzero scalars such that x x x x 0 The latter equation is equivalent to . Performing elementary row operations, We end up with an equivalent linear system Let . Then and x Therefore we find that x x 0 11. The matrix containing the given vectors as columns, X , is of size . Since ~ , we can augment the matrix with rows of zeros. The resulting matrix, X , ~ is of size . Since X is square matrix, with at least one row of zeros, it follows ~ ~ that X Hence the column vectors of X are linearly dependent. That is, there ~ is a nonzero vector, c , such that X c 0 . If we write only the first rows of the latter equation, we have X c 0 . Therefore the column vectors of X are linearly dependent. 12. By inspection, we find that x Hence x x x x 0 , and the vectors are linearly dependent. 13. Two vectors are linearly dependent if and only if one is a nonzero scalar multiple _________________________________________________________________...
View Full Document

## This note was uploaded on 03/11/2014 for the course MA 303 taught by Professor Staff during the Spring '08 term at Purdue.

Ask a homework question - tutors are online