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Study Guide for Second Exam - Differential Equation and Linear Algebra | MA 221, Exams of Mathematics

Material Type: Exam; Class: Diff Equations & Matrix Alg I; Subject: Mathematics; University: Rose-Hulman Institute of Technology; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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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

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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 s does the following system has non-trivial solutions?  

s 5 s

x y z

  1. 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 Xp is a particular solution and Xh is a homogeneous solution. E.g., worksheet #
    • Understand the rank theorem.
  2. 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.
  3. 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. Curve fitting
    • Be able to set up and solve the and interpolation problem that may involve derivatives. E.g., p. 225, 1-10, 27-
    • Be able to find the least squares fit of a polynomial to univariate data.
  5. Second order DE′s,
    • 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.