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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 Spring '08
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 Linear Algebra, eigenvector

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