MA20008 Algebra 1, 2004, Sheet 9

  Geoff Smith, http://www.bath.ac.uk/∼masgcs

1. Let U be the set of polynomials in the variable X with coefficients in ℝ. We define an inner product ⟨ , ⟩ on U via

⟨f, h⟩ = ∫_{0}^{1} fh dX.

Thus U is a vector space of ℝ in the natural way. Let V be the subspace of U consisting of polynomials of degree at most 3. Given the basis 1, X, X^{2}, X^{3}, run the Gram-Schmidt algorithm to produce an orthonormal basis of V.

2. Let V be an inner product space with orthonormal basis v_{1}, v_{2}, …, v_{n}. If the Gram-Schmidt algorithm is used to modify this basis, what is the output?

3. Let V be an inner product space with orthonormal basis v_{1}, v_{2}, …, v_{n}. We obtain another basis v_{n}, v_{n−1}, …, v_{1} by reversing the order of the vectors. Run the Gram-Schmidt algorithm on each of these bases in turn. Is it true that the output orthonormal bases are the reverse of each other?

4. For r = 0, 1, 2 define functions f_{r} : ℝ ⟶ ℝ by f_{r} : θ ↦ cos rθ for all real numbers θ. The collection of all functions f : ℝ ⟶ ℝ has a natural vector space structure. Let V be the subspace spanned by f_{0}, f_{1}, f_{2}. Define an inner product on V via ⟨f, h⟩ = \frac{1}{2π} ∫_{0}^{2π} fh dθ. Run Gram-Schmidt to obtain an orthonormal basis of V.

5. Suppose that e_{1}, e_{2}, …, e_{n} is an orthonormal basis of the inner product space V. Let U_{r} = ⟨e_{1}, e_{2}, …, e_{r}⟩. Suppose that v ∈ V. Show that among all vectors x ∈ U_{r}, the one which minimizes ||v − x|| is ∑_{i=1}^{r}⟨v, e_{i}⟩e_{i}.

6. Suppose that V is an inner product space of dimension n, and α : V ⟶ V is a linear map. Suppose that α carries some orthonormal basis to an orthonormal basis. Show that α carries each orthonormal basis to an orthonormal basis.