Linear Algebra
Study Guide for Exam 1
I really suggest that you make flashcards for all the important definitions in this course.
There is a lot of new terminology and you need to know what I (or the problem) is asking
you to do. This class is hard for students when they do not memorize the definitions. I
will ask you to define various terms and answer True/False questions based on the theory
(similar to your homework). You may use a graphing calculator not capable of symbolic
manipulations. No cell phones.
Section 1.1 You need to know the definitions for or how to do/use the following:
 Linear system
 Solution to a linear system
 Equivalent linear systems
 Consistent linear systems
 Inconsistent linear systems
 Elementary row operations
 Existence vs. uniqueness of a solution
 Row reduce an augmented matrix to echelon form (REF)
Section 1.2 You need to know the definitions for or how to do/use the following:
 Definition of Row Echelon form (REF) and Reduced Row Echelon Form (RREF)
 Theorem 1
 Pivot position and pivot column
 Row reduction algorithm to REF and RREF
 Basic vs. free variables
 Parametric descriptions of solution sets
 Back Substitution
 Theorem 2
Section 1.3 You need to know the definitions for or how to do/use the following:
 Definition of a vector, equality of vectors, vector addition, scalar multiplication
 Geometric descriptions of vectors in R2 and R3
 Parallelogram rule for vector addition
 Vectors in Rn
 Linear combinations
 Algebraic properties of Rn
 Solving vector equations using augmented matrices
 Definition of a spanning set
 Applications of linear combinations
Section 1.4 You need to know the definitions for or how to do/use the following:
 Definition to multiply Ax on page 41.
 Theorem 3
 Existence of solutions to Ax=b
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Theorem 4!!!!!!!!!
Row-vector rule for computing Ax
Theorem 5
Section 1.5 You need to know the definitions for or how to do/use the following:
 Homogeneous linear systems
 Trivial solutions vs. nontrivial solutions
 How to determine if a homogeneous system has a nontrivial solution
 Parametric vector form of the solution set
 Theorem 6; understand the relationship between the solution to Ax=0 and Ax=b
Section 1.6
Know how to do the assigned homework problems.
Section 1.7 You need to know the definitions for or how to do/use the following:
 Definition of linear independence – word for word!
 Definition of linear dependence
 How to tell if the columns of A are linearly independent
 Linear independence relationships with a set of one or two vectors.
 Theorem 7
 Theorem 8
 Theorem 9
Section 1.8 You need to know the definitions for or how to do/use the following:
 Definition of a transformation from Rn to Rm
 Domain, co-domain and range.
 Matrix transformations
 Definition of a linear transformation (very important!)
 Equations 3 and 4 on page 77
 Be comfortable with all the new notation as well as terminology in this section.
Section 1.9 You need to know the definitions for or how to do/use the following:
 Theorem 10 – very important!!!!!!!
 Have a good understanding of tables 1, 2 and 3 on pp. 85-87.
 Definition of an onto mapping
 Definition of a 1-1 (one to one) mapping
 Theorem 11
 Theorem 12
Section 1.10 Anything out of this section would be put on a take-home part of the
exam. I may not even do that.
To study for the exam, know the above list of topics. Do your homework. Review your
quizzes. Do the supplementary exercises on pp. 88-90 (odds are in the back of the book).