late-night-linear-algebra

If in your focus to complete your linear algebra homework you forgot to post a poll question, which linear algebra question would be the best to ask your poll recipients?

• Given any unit vector u of length n over the complex numbers, define Hu = I - 2uu*. Prove Hu is self-adjoint and unitary.~
• Show that the infinite set of functions corresponding to a basis of the set of trigonometric polynomials is orthonormal in the set of all continuous functions in [0, 2pi].~
• In each case below, V is an inner product space over F, and W ≤ V is a finite-dimensional subspace. Find (with proof) a nice description of W-perp, and a nice formula for orthogonal projection onto W. Note: These instructions are intentionally vague. I want you to find and state a clear description of the truth, with proof.~
• Consider the function f(t) = e^sin(t). Plot f on a set of axes together with the nearest trigonometric polynomial of degree at most N for N = 1, 3, 5.~
• Consider the function f(t) = e^sin(pi*t). Given N, let gN be the (honest) polynomial of degree at most N that is closest to f in the set of all continuous functions over [-1, 1]. Plot f together with gN for N = 1, 3, 5.~