### Study Guide for Test 3 - Ch 1-4, 7, Computer Graphics
and Related LAMP Materials

**1 8.5 x 11 sheet with writing on both sides allowed.
You may put anything you want that fits on your sheet.**
Calculator allowed.
The test will be taken in the computer lab so that everyone has access to
Maple.

**Be sure to study** tests 1 and 2 and any related material
that you need to brush up on (test 3 will be comprehensive),
PS 6 solutions, and the solutions to Lamp problems
Chapter 5 Module 1 probs 1 and 8, and Chapter 6 Module 4 problems 7
and 8, which are up on WebCT.

In addition to knowing how to set problems up and
perform the relevant by-hand and/or Maple/calculator
calculations you need to be able to
quickly multiply matrices, take determinants, set up and
solve systems of equations by-hand and on Maple using Gaussian elimination
and the inverse method,
be able to find eigenvalues,
eigenvectors and the matrix P and P inverse (if it exists),
produce various transformation matrices and use them,
and you also need to know examples and counterexamples.

From the WebCT solutions and your old tests, in addition to reviewing them
in general, also be sure to carefully go over
PS 4 and PS 5 Lamp cement problems,
PS 6 rotation matrix eigenvalue problem,
PS 6 fox problem including the extra credit portion,
Chapter 6 Module 4 Problem 7 on projection matrices,
and Problem Set and Test problems where a matrix had an unknown value
such as k for some entry
(Be sure you know how to solve these problems by doing Gaussian elimination
to be able to determine how many solutions you have for various k).
You can expect to see problems which are similar to these, and can
also expect to have a problems which ask you to give examples,
such as examples of matrices which have
a certain number of eigenvalues and eigenvectors.

**I advise you to include the following onto your cheat sheet:**
Summary of Equivalent Conditions for Square Matrices on
p. 221 (don't worry about Rank as we did not cover this.)

the 5 general 2D transformations
in matrix form from LAMP module 4 section 1

general projection matrix (onto a line through the origin
theta degrees away from the x-axis),

general reflection matrix

general rotation matrix

general dilation matrix

general shear matrix

the three 3-D rotation forms at the bottom of page 370

vector space axioms 1 and 6 and their negations (ie the subspace axioms),

definition of basis, linear independence and span

definition of eigenvalue and eigenvector

Aside from v.s. axioms 1 and 6,
you do NOT need the other vector space axioms or their negations.

**The Point of Our Work**
You need to know some big picture ideas
- applications of matrices to real life, and the general
purpose and usefulness of various things, such as recent Lamp modules,
that we have done in the class.

**Proofs** Know the following two proofs:

7.1 p. 391 number 36 (we did this in class and gave more explanation
than the book) -
Prove that lambda = 0 is an eigenvalue of A if and only if A is singular.

6.1 p. 331 number 36 (we did this in class and gave more
explanation than the book) -
Prove that the general rotation matrix gives rise to a linear transformation
from R^2 to R^2 which has the property that it rotates every vector in
R^2 counterclockwise about the origin through the angle theta.

**Maple Commands** Be sure that you know how to input,
read and use the output given to you from Maple on the following:

with(LinearAlgebra): with(Lamp):

Defining a Matrix M:=Matrix([[1,... For example, M:=Matrix([[1,2,3],[4,5,6],[1,0,1]]);

Reduce(M, form=rref);

Evectors(M);

Inverse(M);

Det(A);

A.B;

Inverse(P).A.P;