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June 25, 2016
Posted by **Neeraj** on Thursday, August 11, 2011 at 7:59am.

- Linear algebra -
**MathMate**, Thursday, August 11, 2011 at 9:01amThree vectors are linearly independent if the determinant formed by the vectors (in columns) is non-zero.

So for u=(1,-1,-1), v=(a,b,c), w=(d,e,f)

There are many possible choices of v and w such that the determinant

1 a d

-1 b e

-1 c f

is non-zero.

The simplest way is to create a triangular matrix such that the diagonal is all non-zero, or

1 0 0

-1 b 0

-1 c f

where b and f are non-zero, and the determinant evaluates to b*f≠0.

Example: (b=-1,c=1,f=1)

(1,-1,-1),(0,-1,1),(0,0,1) are linearly independent because the determinant

1 0 0

-1 -1 0

-1 1 1

evaluates to 1*(-1)*(-1)=-1 ≠ 0

Note:

vectors that are orthogonal to each other are linearly independent, since each cannot be a linear combination of the others.

However, linear independent vectors need not be orthogonal. Therefore their dot products need not be zero.