Math 317: Linear Algebra
Homework 3
Due: February 9, 2015
The following problems are for additional practice and are not to be turned in: (All
problems come from Linear Algebra: A Geometric Approach, 2nd Edition by ShifrinAdams.)
Exercises: Section 2.1: 1–5, 7, 8, 14
Section 2.2: 1–7, 10, 11
Turn in the following problems.
1. Divide the interval [0, 1] into five equal subintervals, and apply the finite difference method in order to approximate the solution of the two-point boundary value
problem
y 00 (t) = 125t,
y(0) = y(1) = 0,
at the four interior grid points. Compare your approximate values at the grid points
with the exact solution at the grid points. Note: Typically, you should not expect
very accurate approximations with only four interior grid points. (This is a special
(t3 − t).
problem! :-))The exact solution is 125
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2. Divide [0, 1] into n + 1 equal subintervals and apply the finite difference approximation method to derive the linear system associated with the two-point boundary
value problem
y 00 (t) − y 0 (t) = f (t),
y(0) = y(1) = 0.
Hint: To approximate y 0 (t), I suggest using the following to make your calculations
a bit easier:
y 0 (t) ≈
y(t + h) − y(t − h)
.
2h
This is a centered approximation for y 0 (t) (much like the centered approximation we
came up with for y 00 (t) in class).
3. Section 2.1, Problem 6
4. Section 2.1, Problem 10
5. Section 2.1, Problem 11
6. (a) Suppose that A and B are two n × n upper triangular matrices. Prove that
the product AB is upper triangular.
(b) If A and B are two n × n upper triangular matrices, what are the diagonal
entries of AB?
(c) Is it true that the product of two n × n lower triangular matrices is again lower
triangular? Justify your answer.
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