Suppose that A is a matrix of real numbers Provo that any ve

Suppose that A is a matrix of real numbers. Provo that any vector in the nullspace of A is orthogonal to any vector in the nullspace of A.

Solution

Suppose that A is a matrix of real numbers. Prove that any vector in the nullspace of A is orthogonal to any vector in the nullspace A.

     if we take finite dimensional vector space V and subspaces W and X of V, we have that W^T=X^T if and only if W=X.

for this One implication is trivial

   For the other, suppose W^T=X^TTake xX.

and we know such that x=w+w, wW, and wW^T=X^T. Then

   ,0=xw=(w+w)w=ww+ww=ww,

so w=0,w=0, whence x=wW, and so XW. By symmetrical arguments, we likewise have WX, so W=X.

both W,X are null space vectors.