Linear Algebra -- Spring 2017

Information about the first exam

The first exam will be Friday, 2/17/17.  For this exam you are responsible for knowing the material in Chapter 1 of the text, sections 1-5 and 7, plus the handout on analyzing column dependencies, and Chapter 2 sections 1-3. 

As a general rule, I try to make each of my exams cover a variety of different kinds of knowledge.  You certainly need to understand and know how to apply procedures.  For example, be able to solve a system of simultaneous equations, row reduce a matrix, formulate the vector parametric form for the general solution of a system, and determine whether a set of vectors is independent.

You also need to have a factual understanding of what is true.  That means knowing what the definitions and theorems say, both in a conceptual way, as well as a technically correct formulation.  Otherwise, if you have a fuzzy understanding of the definitions, you will be in  no position to understand the theorems and techniques that make reference to them.  Similarly, if you don't know the statements of the theorems, you cannot say for sure, in any particular situation, whether or not the theorems can be applied.  For example, one theorem states that a matrix equation Ax = b will definitely be consistent if there is a pivot in every row.  But does that mean every row of the coefficient matrix A or the augmented matrix [A|b]?  As a matter of fact, it refers to A, and if you apply it to the augmented matrix, you can be led to an incorrect conclusion.  So you need to know technically correct statements of definitions and theorems, and should be prepared to see some items on the exam that will ask you to demonstrate that knowledge.

Another kind of exam question tries to get at conceptual understanding beyond the exact statements of theorems and definitions.  For example, the definition of linear independence is formulated in terms of nontrivial solutions to a certain homogeneous equation.  But you should also understand conceptually that a set of vectors is independent when no one of the vectors can be expressed as a linear combination of the others.  So, in addition to knowing the correct mathematical statements of definitions and theorems, you should also be able to explain, in your own words, what these things mean.  Alternatively, you may be asked to answer true false or multiple choice questions and reason about specific situations, just as in the homework assignments.

For a sample exam,  CLICK HERE.  Be advised that this is not a comprehensive review.  It serves merely to illustrate a mix of question styles (procedures, conceptual understanding, definitions and theorems) and the approximate length you can expect.  The sample exam only covers a subset of the topics that might appear on the real exam.  For this reason it is important that you systematically review ALL of the material we have covered since the start of the course.  In terms of length, difficulty, and style of questions, the sample exam should be a pretty good guide to what our exam will be like.