Math 207 Fall 2014  Problem Set 1

Problem 1. The set B = {1 + x, 1 − x, 1 − 2x^{2}, 1 + x + x^{3}, x^{2} − x^{4}} is a basis for V, the vector space of polynomials of degree less than or equal to four. Using the ordered basis B, there is a correspondence

V ↔ ℝ^{5}
p(x) ↔_{B} (p(x))

(a) What polynomials in V correspond to the vectors (\begin{matrix}1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}), (\begin{matrix}1 \\ 0 \\ 0 \\ 2 \\ 0 \end{matrix}), and (\begin{matrix}1 \\ 0 \\ −2 \\ −1 \\ 1 \end{matrix}) in ℝ^{5}?

(b) What vectors in ℝ^{5} corresponds to the polynomials x^{4} −8, x^{4} +1, 2x^{4} −7, and 4x^{4} −14 in V?

(c) By replacing the polynomial 1 + x + x^{3} with the polynomial x^{3}, we can obtain a new basis B′ = {1 + x, 1 − x, 1 − 2x^{2}, x^{3}, x^{2} − x^{4}}. Find

B′(x^{4} − 8), _{B′}(−x^{4} − x^{3} + 5x^{2} − 1), and _{B′}(−x^{3} + 5x^{2} − 9).

Problem 2. Each of the pictures below shows a polynomial approximation for cos(2t). Five of them are degree four polynomials obtained by projection onto the subspace of quartic polynomials, using one of the following inner products.

⟨f, g⟩_{1} = f(2)g(2) + f′(2)g′(2) + f′′(2)g′′(2) + f′′′(2)g′′′(2)
⟨f, g⟩_{2} = ∫_{0}^{4} f(t)g(t)dt
⟨f, g⟩_{3} = f(0)g(0) + f(1)g(1) + f(2)g(2) + f(3)g(3) + f(4)g(4)
⟨f, g⟩_{4} = f(0)g(0) + f(2)g(2) + f′(2)g′(2) + f′′(2)g′′(2) + f(4)g(4)
⟨f, g⟩_{5} = f(0)g(0) + f′(0)g′(0) + f(2)g(2) + f(4)g(4) + f′(4)g′(4)

The sixth approximation is obtained by gluing together two degree two polynomials defined on [0, 2] and [2, 4], each determined by projection using, respectively, the inner products

⟨f, g⟩_{6} = f(0)g(0) + f(2)g(2) + f′(2)g′(2)
⟨f, g⟩_{7} = f(2)g(2) + f′(2)g′(2) + f(4)g(4)

Math 207 Fall 2014  Problem Set 1

Problem 2. (Continued)

(a) Which picture goes with which inner product?

(b) Explicitly compute the approximation corresponding to ⟨ , ⟩_{5}. Problem 3. Let V be the space of polynomials of degree less than or equal to three and let

⟨p, q⟩ = p(0)q(0) + p′(0)q′(0) + p(1)q(1) + p′(1)q′(1).

(a) Prove that ⟨p, q⟩ defines an inner product on V. In particular, prove that if ⟨p, p⟩ = 0 then p = 0.

(b) Compute the angle between the polynomials 2t^{3} − 3t^{2} and 1.

(c) Use Gram-Schmidt to replace the basis {1, t, t^{2}, t^{3}} by an orthogonormal set.

(d) The following polynomials form a very nice basis for V. e_{1} = x(x − 1)^{2}

e_{2} = x^{2}(x − 1)
e_{3} = (2x + 1)(x − 1)^{2}
e_{4} = x^{2}(1 − 2(x − 1))

Projecting an arbitrary differentiable function f onto V yields a linear combination of the basis vectors

a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + a_{4}e_{4}.

Describe (simply!) the coefficients a_{i} in terms of the function f.