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(i) They are linearly dependent since the 2nd vector is twice the 1st vector.
All 3 vectors lie in the x-z plane, so they don't span 3D space.
(ii) They are linearly independent.
Note that the cross-product of the first two is (-1,1,1).
If the third vector is not perpendicular to the above cross-product,
then the third vector does not lie in the plane defined by the first two vectors.
(-1,1,1) "dot" (1,1,-1) = -1+1-1 = -1, not zero, so 3rd vector is not perpendicular
to the cross product of the other two.
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If a set of vectors spans R3 then the set is linearly independent?
No it is not. It's possible to have to have a set of vectors that are linearly dependent but still Span R^3. Same holds true for reverse. Linear Independence does not guarantee Span R^3. IF both conditions are met then that set of vectors is called the Basis for R^3. So, for a set of vectors, S, to be a Basis it must be:(1) Linearly Independent(2) Span S = R^3.This means that both conditions are independent.
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