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:

- Computations;
- Statements of definitions and theorems;
- Proofs;
- 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:

- Every system of linear equations is consistent.

False. The linear systemYour reason might also be a more general principle:0=1(augmented matrix[0 | 1]) has no solutions, and is therefore inconsistent.

False. The linear systemEither way, your answer should be as specific as possible to ensure full credit.Ais consistent only if the RREF ofx=b[A |has no row in which the only nonzero entry lies in the last column.b]

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:

You should also know the elementary row operations, but not necessarily by number.

1.1 matrix scalar size of a matrix m×nmatrixsquare matrix (i,j)-entrysubmatrix transpose row column a_{ij}a_{i}matrix sum A+Bscalar multiple cAnegative -Asubtraction A-Bzero matrix vector components of a vector n-tuple1.2 linear combination coefficients of a lin comb standard vectors of R^{n}matrix-vector product Avidentity matrix I_{n}rotation matrix A_{theta}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

**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
? Is**R**^{n}*A*consistent for all**x**=**b**?**b**in**R**^{n}

**Good luck.**

hsu@mathcs.sjsu.edu