Format and topics for exam 1
Math 129a

General information. Exam 1 will be a timed test of 50 minutes, covering section 1.1-1.4, pp. 54-55, and 1.6 of the text. No books, notes, calculators, etc., are allowed. Most of the exam will rely on understanding the problem sets (including problems to be done but not to be turned in), the quizzes, and the definitions and theorems that lie behind them. If you can do all of the homework and the quizzes, and you know and understand all of the definitions and the statements of all of the theorems we've studied, you should be in good shape.

You should not spend time memorizing proofs of theorems from the book, but you should defintely spend time memorizing the statements of the important theorems in the text.

Types of questions. In general, there are four types of questions that will appear on exams:

  1. Computations;
  2. Statements of definitions and theorems;
  3. Proofs;
  4. True/false with justification.

Computations. These will be drawn from computations of the type you've done on the problem sets. You do not need to explain your answer on a computational problem, but show all your work.

Statements of definitions and theorems. In these questions, you will be asked to recite a definition or the statement of a theorem from the book. You will not be asked to recite the proofs of any theorems from the book.

Proofs. These will resemble the questions that have been assigned on paragraph homework. You should answer in complete sentences, if you have time, but you won't have to write a lot to answer any given question; to be more precise, you shouldn't have to write more than a few sentences to answer any given question.

True/false with justification. This type of question may be less familiar. You are given a statement, such as:

If the statement is true, all you have to do is write "True". (However, see below.) If the statement is false (like the one above), not only do you have to write "False", but also, you must give a reason why the statement is false. Your reason might be a very specific counterexample:
False. The linear system 0=1 (augmented matrix [0 | 1]) has no solutions, and is therefore inconsistent.
Your reason might also be a more general principle:
False. The linear system Ax=b is consistent only if the RREF of [A | b] has no row in which the only nonzero entry lies in the last column.
Either way, your answer should be as specific as possible to ensure full credit.

Depending on the problem, some partial credit may be given if you write "False" but provide no justification, or if you write "False" but provide insufficient or incorrect justification. Partial credit may also be given if you write "True" for a false statement, but provide some kind of partially reasonable justification. (In other words, it can't hurt to try to justify "True" answers, and it can help you.)

If I can't tell whether you wrote "True" or "False", you will receive no credit. In particular, please do not just write "T" or "F", as you may not receive any credit.

Not on exam. Leontief models will not be on the exam.

Definitions. The most important definitions and symbols we have covered are:

1.1 matrix scalar
size of a matrix m×n matrix
square matrix (i,j)-entry
submatrix transpose
row column
aij ai
matrix sum A+B scalar multiple cA
negative -A subtraction A-B
zero matrix vector
components of a vector n-tuple
1.2 linear combination coefficients of a lin comb
standard vectors of Rn matrix-vector product Av
identity matrix In rotation matrix Atheta
1.3 linear equation coefficients of a lin eqn
constant term of a lin eqn system of linear eqns
solution to a lin sys solution set of a lin sys
consistent inconsistent
equivalent lin systems coefficient matrix of a lin sys
augmented matrix of a lin sys elementary row operations
leading entry of a row row echelon form
reduced row echelon form basic variables
free variables general solution to a lin sys
"the" RREF of a matrix
1.4 pivot position pivot column
rank nullity
1.5 Kirchoff's voltage law Kirchoff's current law
1.6 span (noun), to span (verb) spaninng set
You should also know the elementary row operations, but not necessarily by number.

Theorems, results, algorithms. The most important theorems, results, and algorithms we have covered are listed below. You should understand all of these results, and you should be able to cite them as needed.

Sect. 1.1:
Thm. 1.1 (arithmetic of matrix addition and scalar multiplication).
Sect. 1.2:
Thm. 1.2 (arithmetic of matrix-vector multiplication and scalar multiplication).
Sect. 1.3:
Thm. 1.3 (every matrix has a unique RREF).
Sect. 1.4:
Gaussian elimination (steps 1-6), Thm. 1.4 (tests for consistency).
Sect. 1.6:
The Fat Matrix Theorem (Thm. 1.5), properties of span (Thm. 1.6).

Types of computational problems. You should also know how to do the following general types of computations. (Note also that on the actual exam, there will be problems that are not of these types. Nevertheless, it will be helpful to know how to do all these types.)

Sect. 1.1:
Adding matrices; scalar multiples of matrices; transpose of a matrix.
Sect. 1.2:
Linear combinations; matrix-vector products.
Sect. 1.3:
Solving linear systems in RREF; putting solution sets in vector form. When is a linear system consistent? When does a linear system have 0,1,infty solutions?
Sect. 1.4:
Gaussian elimination.
Sect. 1.6:
Does a given set of vectors span Rn? Is Ax=b consistent for all b in Rn?

Good luck.