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a)       Find a linearly independent set of vectors that spans the same subspace...

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hockeyfan54 | Student | (Level 1) Salutatorian

Posted March 19, 2013 at 4:16 PM via web

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a)       Find a linearly independent set of vectors that spans the same subspace of R^3 as that spanned by the vectors

Vector u = [[-3],[-3],[3]]

Vector v [[-1],[-1],[-3]]

Vector w = [[2],[2],[0]]

b) Write vector w as a linear combination of vector u and vector v .

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pramodpandey | College Teacher | (Level 3) Valedictorian

Posted March 20, 2013 at 10:27 AM (Answer #1)

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Find the a basis for the set of vectors.So  form a matrix, with each row being formed by a vector.

`[[-3,-3,3],[-1,-1,-3],[2,2,0]]`

Reduce it to row echelon form ( May be by Gaussian elimination) as.

`[[1,1,0],[0,0,1],[0,0,0]]`

This means that only two of the vectors in  set are linearly independent i.e. u and v vectors alone will span the same subspace of R^3  as all three vectors.

w=au+bv

2=-3a-b

2=-3a-b

0=3a-3b

solving this sytem we get a=b=(-1/2)

W= (-1/2)u+(-1/2)v

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