find the basis of annihilator of subspace W in R^{5}, which is spanned by (2,-2,3,4,1), (0,0,-1,-2,3), (-1,1,2,5,2), (1,-1,2,3,0)?
please explain
i have reduced to the echo-matrix then i don't have the idea.
Use the Gauss–Jordan elimination
1) reverse rows 1 and 3;
2) add row 1 to row 3, multiply row 1 by –2 and add row 4;
3) multiply row 2: by 4 and add to row 3, by –1 and add row 3 to row 1;
4) devide row 3 by 14 and row 4 by –2;
5) subtract row 3 from row 4;
6) multiply row 3 by –3 and add to row 2;
7) multiply row 2 by 2 and add to row 1
So, $\displaystyle (1,1,0,0,0)$ and $\displaystyle (1,0,-2,1,0)$.
by inspecting the rank of the matrix DeMath has produced, we see that dim(W) = 3, hence dim(ann(W)) = 2. the vectors DeMath has produced aren't elements of (R^{5})*, unless you interpret them as the coordinates of the basis elements of ann(W) in the dual basis to the standard basis of R_{5}.
that is to say, there is a natural isomorphism between ann(W) and W^{⊥}:
f_{x} given by f_{x}(y) = <x,y> is in ann(W) iff x is in W^{⊥}.
the chief difference is: ann(W) is a subspace of the dual space, whereas W^{⊥} is a subspace of the original vector space.