Bonus Assignment

This optional assignment is worth some extra points, the number of which will be determined by the quality of what you turn in.

There is no penalty for not doing this assignment.

All write ups for this assignment must be typed or written very neatly by hand. It might help you to use a graphing calculator or computer algebra system such as Mathematica, Wolfram Alpha, Sage, or Maple.

Recall \(V = C([-1,1])\) is the vector space of continuous real valued functions with domain \([-1,1]\). Define an inner product on \(V\) by \(\langle f(x),g(x) \rangle = \int_{-1}^1 f(t)g(t)\,dt\) for all \(f(x),g(x)\in V\).

Here are some possible topics to explore (you may find other interesting things to include in your write up):

  1. Starting with \(\{1,x,x^2,x^3\}\), use Graham-Schmidt to find an orthonormal basis for the subspace \(P_3(\mathbb{R})\) of \(V\).
  2. Find the element of \(P_3(\mathbb{R})\) nearest to \(f(x) = e^x\) and to \(g(x) = \cos(x)\). Nearest in the sense of norm/length induced by the given inner product.
  3. Recall that the Taylor polynomial of degree 3 (which is an element of \(P_3(\mathbb{R})\)) is a cubic polynomial that you learned about in calculus. Taylor polynomials approximate functions. Find the Taylor polynomials of degree 3 for \(f(x) = e^x\) and for \(g(x) = \cos(x)\), centered at the origin.
  4. Compare and contrast the approximations from (2) to the approximations from (3). For example, is the Taylor polynomial a better approximation outside of the interval \([-1,1]\)? If \(T\) denotes the Taylor polynomial and \(Q\) denotes the approximation from (2) (for either \(f\) or \(g\)), how does \(||T – f|| \) compare to \(||Q-f||\)? and \(||T-g||\) to \(||Q-g||\)?
  5. Which type of approximation do you think is better?

Homework 8

Due Wednesday, October 30, in class

The starred (*ed) problems are recommended only, i.e. not for credit.

Section 3.3 problems 1, 2bd, 3bd, 4a
Section 4.2 problems 1*, 11, 27*
Section 4.3 problem 22*
Section 5.1 problem 1, 3d, 4c, 4e*, 7, 17

Homework 5

Due Wednesday, October 2, in class

Section 2.2 problems 1, 4, 14, 15, 16
Section 2.3 problems 1, 3, 10, 16

Then

A. Let \(V,W\) be vector spaces over \(F\) with ordered bases \(\beta = \{v_1,\ldots, v_n\}\) and \(\gamma = \{ w_1,\ldots, w_m\}\), respectively. Show that the function \(G:\ \mathcal{L}(V,W) \to M_{m\times n}(F)\) given by \(G(T) = [T]_\beta^\gamma\) is a linear transformation. (Side note: \(G\) is an element of \(\mathcal{L}(\mathcal{L}(V,W), M_{m\times n}(F)).\)

Homework 4

Due Wednesday, September 25, in class

Section 1.7 problems 1, 3*, 5, 6
Section 2.1 problems 1, 5, 6, 10, 14, 35

and

A. Use Theorem 2.6 to prove this problem. If \(V\) is a finite dimensional vector space over \(F\) of dimension \(n\), then there is a one-to-one and onto linear transformation \(T:\ V \to F^n\).

* Hint to the book’s hint (or simply a hint on its own): The idea is to find an infinite set that is linearly independent. Then carefully argue why an infinite linearly independent set implies \(V\) is infinite dimensional. There are such linearly independent sets that do not use the transcendence of \(\pi\), if you don’t want to follow the book’s hint.