SOLUTION: Let x and y be vectors in R^4 with {x,y} being linearly independent. Prove that there exists a non zero vector z that is orthogonal to both x and y. Thanks!

Algebra ->  Proofs -> SOLUTION: Let x and y be vectors in R^4 with {x,y} being linearly independent. Prove that there exists a non zero vector z that is orthogonal to both x and y. Thanks!      Log On


   

Question 1035834: Let x and y be vectors in R^4 with {x,y} being linearly independent. Prove that there exists a non zero vector z that is orthogonal to both x and y.
Thanks!
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
z is orthogonal to x iff . Similarly with z and y.

Let and . Hence we wish to find a vector such that both are true:



This is a system of two equations with four variables, and one can find a nonzero solution z that works (this follows from the rank-nullity theorem in linear algebra, since the matrix whose entries are x_1, ..., x_4 \\ y_1, ..., y_4 has non-zero nullity, so the nullspace contains some nonzero vector which we can choose for z).