Give an orthonormal basis for null(T), where T∈L(C4) is the map

| August 30, 2017

estion
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 finite-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 satisfies

dim(U⊥)=dim(V)−dim(U).

7. Let V be a finite-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 satisfies 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)

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