MAS 3105
Review - Exam 1
September 16, 2010
When: September 23, 2010, in class
What Material: Sections 1.1-1.4, 1.6-1.7, 2.1, 2.3-2.4, 2.7-2.8.
Procedure: The exam will be closed book. You will be allowed one sheet of notes, 8-1/2
by 11 inches, front and back. You may not use any graphing calculators of any kind. You
may bring in a basic calculator if you want.
How to Study: The exam will have straightforward computation questions (find the
inverse of a matrix, solve a system of equations, determine if a vector is in the span of a
set, etc) and the exam will have “understanding” questions. These are questions that test
whether or not you understand the definitions and theorems from the class. All the
problems from the homework that were at the end of a section – those were
“understanding” questions.
You WILL be expected to show your work on all problems. That means you need to
show all steps in a Gaussian elimination.
You should run through all of the True/False questions. These really help you with
definitions and theorems.
Computations are straightforward so far, and everything boils down to finding linear
combinations and/or performing Gaussian elimination. Make sure you can do both tasks
quickly, and make sure you don’t make arithmetic mistakes.
You should go over every homework problem assigned, as well as all problems from the
chapters that are similar. Continue to practice until you can do the problems comfortably
in a test situation (one page of notes and a calculator, no additional notes, no back of the
book, no help from friends). Also focus on quiz problems. If I found a type of problem
important enough for a quiz, it is probably important enough for an exam. Finally go
over the review exercise sets at the end of the chapter. Even though it includes some
material that I did not cover, it is good in that the topics are mixed together, just like on
the exam.
Specific Topics:
 Matrix and Vector Arithmetic: You need to be able to add, subtract, and scalar
multiply both matrices and vectors. You need to be able to do matrix-vector
multiplication, matrix-matrix multiplication, as well as vector linear combinations.
Make sure you know all of the algebra rules (Theorems 1.1, 1.3, 2.1). You need to be
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able to transpose a matrix, and you need to understand the definitions associated with
transposes (symmetric matrices and skew-symmetric matrices). Also know how
transpose and arithmetic interact (Theorems 1.2 and 2.1-g). Computation questions:
1.1: 1-24, 1.2: 1-16, 2.1: 1-32. Understanding questions: 1.1: 57-80, 1.2: 80-89, 2.1:
54-57, 59-64, 67, 68, 70.
Systems of Linear Equations: Given a system of linear equations, determine if the
system is consistent. If so, find the solution set. For this (as well as many other
problems), you need to find the reduced row echelon form of a matrix, and you will
do that using Gaussian elimination. Computation questions: 1.3: 1-54, 1.4: 1-42 (1734 especially). Understanding \questions: 1.3: 78-86, 1.4: 73-93.
Span: Given a set of vectors, determine if another vector is in the span of the set.
Also, given a set of vectors, determine if they generate all of Rm. In both cases, this
requires Gaussian elimination. Make sure you know Theorem 1.6 – it can save you a
bunch of work. Computation questions: 1.6: 1-36 (17-20 especially). Understanding
questions: 1.6: 65-76.
Linear Independence: Given a set of vectors, determine if the set is linearly
dependent or linearly independent. Again, it all boils down to Gaussian elimination.
And again, make sure you know Theorem 1.8. Computation questions: 1.7: 1-50
(especially 39-50). Understanding questions: 1.7: 83-89.
Invertible Matrices: Given a square matrix, find its inverse, or determine if no
inverse exists. Given an elementary row operation, find the corresponding
elementary matrix. Make sure you know how inverses interact with other things
we’ve learned (Theorem 2.2), and make sure you know all of the equivalent
conditions for a matrix to be invertible (Theorem 2.6). Computation questions: 2.3:
9-32, 2.4: 1-18. Understanding questions: 2.3: 56, 58-65, 2.4: 64-73, 89-91.
Linear Transformations: Given a linear transformation, write down its standard
matrix, determine if it is one-to-one, determine if it is onto, and find its nullspace.
Given some information about a linear transformation (like the image of a couple of
vectors), determine the linear transformation. Make sure you know how linear
transformations interact with linear combinations (Theorems 2.7 and 2.8), and make
sure you know how to tell if a map is one-to-one or onto (Theorems 2.10 and 2.11).
Computation questions: 2.7: 1-34, 56-71, 2.8: 1-40, 63-68. Understanding questions:
2.7: 87-100.