Linear algebra
posted by Neeraj .
Find two vectors v and w such that the three vectors u = (1,1,1), v and w are linearly independent independent.

Three vectors are linearly independent if the determinant formed by the vectors (in columns) is nonzero.
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 nonzero.
The simplest way is to create a triangular matrix such that the diagonal is all nonzero, or
1 0 0
1 b 0
1 c f
where b and f are nonzero, 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.