Question

let v=(1,9,-1,5). find a basis of the subspace of R^4 of all vectors perpendicular to v.

Answer 1

Perpendicular to v\[ \rightarrow x_1 + 9x_2 -x_3 + x_4 = 0 \rightarrow x_3 = x_1 + 9x_2 + x_4 \]
Then by decomposition we have
\[\rightarrow \{<1,0,1,0>, <0,1,9,0>, <0,0,1,1>\}\]
is a basis of the subspace.