Multivariable Advanced Calculus

3.10. EXERCISES 69becauseid = ∑ jlδ jl w l w jThus letting M be the matrix whose ij th entry is (Rw i , w l ) , det (R) is defined asdet (M) and 3.48 says∑ ) (M T M il = δ jl .jiIt follows1 = det (M) detThus |det (R)| = |det (M)| = 1 as claimed.3.10 Exercisesi(M T ) = det (M) det ( M ) = det (M) det (M) = |det (M)| 2 .1. For u, v vectors in F 3 , define the product, u ∗ v ≡ u 1 v 1 + 2u 2 v 2 + 3u 3 v 3 . Show theaxioms for a dot product all hold for this funny product. Prove|u ∗ v| ≤ (u ∗ u) 1/2 (v ∗ v) 1/2 .2. Suppose you have a real or complex vector space. Can it always be consideredas an inner product space? What does this mean about Schur’s theorem? Hint:Start with a basis and decree the basis is orthonormal. Then define an innerproduct accordingly.[3. Show that (a · b) = 1 4|a + b| 2 − |a − b| 2] .4. Prove from the axioms of the dot product the parallelogram identity, |a + b| 2 +|a − b| 2 = 2 |a| 2 + 2 |b| 2 .5. Suppose f, g are two Darboux Stieltjes integrable functions defined on [0, 1] . Define(f · g) =∫ 10f (x) g (x)dF.Show this dot product satisfies the axioms for the inner product. Explain why theCauchy Schwarz inequality continues to hold in this context and state the CauchySchwarz inequality in terms of integrals. Does the Cauchy Schwarz inequality stillhold if(f · g) =∫ 10f (x) g (x)p (x) dFwhere p (x) is a given nonnegative function? If so, what would it be in terms ofintegrals.6. If A is an n × n matrix considered as an element of L (C n , C n ) by ordinary matrixmultiplication, use the inner product in C n to show that (A ∗ ) ij= A ji . In words,the adjoint is the transpose of the conjugate.7. A symmetric matrix is a real n × n matrix A which satisfies A T = A. Show everysymmetric matrix is self adjoint and that there exists an orthonormal set of realvectors {x 1 , · · · , x n } such thatA = ∑ kλ k x k x k