(a) Let A be an n × n symmetric matrix, and let v be an eigenvector. Prove that its orthogonal complement under the dot product, namely, V^⊥ = {w ∈ Rⁿ | v₁· w = 0}, is an invariant subspace. 

(b) More generally, prove that if W ⊂ Rⁿ is an invariant subspace, then its orthogonal complement W^⊥, is also invariant.