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!

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) (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).
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