(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 R n v 1 w 0 , is an invariant subspace (b) More generally, prove that if W R n is an invariant subspace, then its orthogonal complement W , is also invariant

Question: (a) Let A be an n n symmetric matrix, and let v be an eigenvector. Prove that its orthogonal complement under the dot product,

(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.

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