6. Give an orthonormal basis for null(T), where T∈L(C4) is the map with canonical matrix
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
6. Let V be a ?nite-dimensional inner product space over F, and let U be a subspace of V. Prove that the orthogonal complement U⊥ of Uwith respect to the inner product ?⋅,⋅? on V satis?es
dim(U⊥)=dim(V)-dim(U).
7. Let V be a ?nite-dimensional inner product space over F, and let U be a subspace of V. Prove that U=V if and only if the orthogonal complement U⊥ of U with respect to the inner product ?⋅,⋅? on V satis?es U⊥={0}.
10. Prove or give a counterexample: The Gram-Schmidt process applied to an an orthonormal list of vectors reproduces that list unchanged.
(Prove by induction)