189-251B: Algebra 2 Assignment 8

Due: Wednesday, March 12

1. Let V be the vector space over the reals consisting of polynomials of degree at most n − 1, and let x_{1}, ⋯, x_{n} be n distinct real numbers.

a) Show that the functionals ℓ_{1}, …, ℓ_{n} on V defined by

ℓ_{j}(p(x)) = p(x_{j}) 1 ≤ j ≤ n

form a basis for the dual space V^{∗}.

b) (Polynomial interpolation). Given real numbers y_{1}, …, y_{n}, show that there is a unique polynomial of degree at most n − 1 satisfying p(x_{j}) = y_{j} for all 1 ≤ j ≤ n, and that it is given by the formula

p(x) = p(x_{1})p_{1}(x) + ⋯ + p(x_{n})p_{n}(x),

where (p_{1}(x), …, p_{n}(x)) is the basis of V which is dual to (ℓ_{1}, …, ℓ_{n}), i.e., satisfies

(p_{1}^{∗}, …, p_{n}^{∗}) = (ℓ_{1}, …, ℓ_{n}).

c) Write down a formula for the basis (p_{1}, …, p_{n}).

2. Proposition 7.2.1. of Eyal Goren’s notes shows that there is a natural injective linear transformation from V to V^{∗∗}, the dual of the dual of V. It then shows furthermore that this map is an isomorphism when V is finitedimensional. Read the proof of Proposition 7.2.1. carefully and note the crucial role played by the finite-dimensionality assumption. (You do not need to write anything, yet...)

Now, give an example to show that the natural map V ⟶ V^{∗∗} need not be surjective in general, if V is not assumed to be finite dimensional. (Warning: you might need some form of the axiom of choice...)

3. Let H = R⟨i, j, k⟩ denote the usual ring of quaternions. (Recall that a typical quaternion is an element of the form

z = a + bi + cj + dk, a, b, c, d ∈ R.

Recall the trace and norm functions on H defined by

trace(z) = 2a = z + z̄, norm(z) = a^{2} + b^{2} + c^{2} + d^{2} = zz̄,

where z̄ = a − bi − cj − dk.

Show that H, equipped with the rule

⟨x, y⟩ := trace(xȳ)

is a real inner product space.

4. Note that H can also be viewed as a complex vector space by viewing the elements of the form a + bi as complex numbers and using the multiplication in H to define the scalar multiplication. Is H equipped with the pairing above a Hermitian space? Explain.

5. Let V be the real vector space of real-valued functions on the interval [−π, π] equipped with the inner product

⟨f, g⟩ := ∫_{−π}^{π} f(t)g(t)dt.

Let W be the subspace of functions spanned by f_{0} = 1/\sqrt{2π}, f_{j}(t) := 1/\sqrt{π} cos(jt), with 1 ≤ j ≤ N and g_{j}(t) := 1/\sqrt{π} sin(jt) with 1 ≤ j ≤ N. Show that f_{0}, f_{1}, …, f_{N}, g_{1}, …, g_{N} is an orthonormal basis for W. Given f ∈ V, give a formula for the coefficients of the function

a_{0}f_{0} + a_{1}f_{1} + ⋯ + a_{N}f_{N} + b_{1}g_{1} + ⋯ + b_{N}g_{N}

in W which best approximates f. (These coefficients are called the Fourier coefficients attached to f.) Compute these coefficients in the case of the function f(x) = x.

5. Let V be the space of polynomials of degree ≤ 2 equipped with the inner product ⟨f, g⟩ = ∫_{0}^{1} f(t)g(t)dt. Find an orthonormal basis for V, by applying the Gramm-Schmidt procedure to the ordered basis (1, x, x^{2}).

6. Let S = (x_{1}, x_{2}, …, x_{N}) and (y_{1}, …, y_{N}) be sequences of real numbers. Fix an integer k less than N. In general, there need not be a polynomial p of degree ≤ k such that p(x_{j}) = y_{j} for j = 1, …, N. The next-best thing one might ask for is a polynomial p of degree ≤ k for which the quantity

(p(x_{1}) − y_{1})^{2} + (p(x_{2}) − y_{2})^{2} + ⋯ + (p(x_{N}) − y_{N})^{2}

is minimised. Describe an approach that would produce such a p. In the case k = 1, give a formula for the coefficients a, b of p(x) = ax + b in terms of x_{1}, …, x_{N}, y_{1}, …, y_{N}.