### 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.

• 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;