Study Guide for Second Hour Exam
1. Determinants. Know how to compute them (2 ×2 and 3 ×3) and how
to interpret them. E.g. for what values of sdoes the following system has
non-trivial solutions?
123
064
s5s
x
y
z
=
0
0
0
2. Null spaces and ranges
•Be able to find a basis for the null space and the range of a matrix
A.
•Be able to write the general solution of a system in the form Xp+Xh
where Xpis a particular solution and Xhis a homogeneous solution.
E.g., worksheet #3
•Understand the rank theorem.
3. Linear independence
•Be able to determine when a vector is a linear combination of other
vectors by setting up and solving a system of equations
•Determine if vectors are linearly dependent by setting up an appro-
priate system.
•e.g., the homework problems from 4.1, 4.2, 4.3,4.4
4. Least squares
•Be able to derive the normal equations for the least squares solu-
tion to Y=AX +E, solve for the least squares approximation and
determine the least squares error E. E.g. #1 on the worksheet #4
5. Curve fitting
•Be able to set up and solve the and interpolation problem that may
involve derivatives. E.g., p. 225, 1-10, 27-34
•Be able to find the least squares fit of a polynomial to univariate
data.
6. Second order DE0s,
•Be able to solve a second order D.E. whose characteristic equation
has distinct real roots. Be able to apply initial data and interpret
the solution or its graph. Homework problems from 5.1.
1