Hence x now setting with solution given by x 17 the

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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 _________________________________________________________________...
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