## Some suggestions as to how to prepare for Test 1

First some nuts and bolts:

Please arrange yourselves on test day in nice straight columns, with TWO empty seats separating adjacent columns (people in a linear algebra course learn the difference between a row and a column...). We will not start the test until this seating pattern is achieved. General rule at York: If an invigilator asks you to move before or during a test or exam, you must just move. The invigilator need not give any reason.

Please have photo ID out on the desk top for the invigilator to see when she is making the rounds with the sign-up sheet. (York photo ID and driver's license are each ok.)

In what follows, I have done little more than just go through my own messy notes of class events beginning 11 September. You may find the result to be of use.

Know what is and what is not an elementary row operation. Know the Gauss-Jordan algorithm as described in class:

If the matrix is 0, stop. Otherwise find the furthest left nonzero column and look for the "furthest up" nonzero entry in it. Switch its row with row 1 if it is not in row 1. Do a type II operation to get a leading 1. "Sweep" under that leading 1 using type III op's. Ignore the row and column of the leading 1 just produced, and all columns to the west of it and all rows to the north of it.

If what remains is a zero matrix, or nothing remains, go to the sweeping above part below. Otherwise, look in what is left for the furthest west nonzero column, as above, and proceed as before.

Eventually we finish with sweeping below the staircase of leading 1's. Now start from the furthest east leading 1 and sweep above it using row op's of type III. (The process of getting a leading 1 and sweeping above or below it is what is called "pivoting" on the GJ handout in class.) Continue the sweeping above, moving westward through the columns with leading 1's.

Know what the coeff. matrix and augmented matrix of a linear system are. Know what a solution, and the solution set, of a linear system are. Know what a parametric description of such a set is, and how to write down such a thing in such a way as to keep the chief happy. Be able to write down such a description, given the GJ form of the augm. matrix of a lin. system.

Know what row-echelon form and reduced row-echelon form are. Know what row-equivalence is, and know the facts mentioned in class about uniqueness and non-uniqueness of r-e., r.r-e. forms for a given matrix. Know what the matrix form of a linear system is.

Know various other definitions, and be able and willing to write them down on our test (and of course willing and able to use them in problems:

rank, consistent system, inconsistent system, row matrix, column matrix, square matrix, main diagonal entries, zero matrix, identity matrix, transpose, symmetric matrix, homogeneous system, trivial solution, diagonal matrix, invertible matrix, inverse of an inv'ble matrix, elementary matrix.

Be familiar with the facts and non-facts about matrix algebra. I will not be as picky as on our second assignment about this; the attitude should be that everything "works" except stuff involving commutativity of matrix multiplication.

Come into the test having done all problems to be handed in and all problems not to be handed in, and ready to use similar techniques to solve similar problems or do similar proofs. I will not list the assigned problems again here; everyone has done what the course outline says and therefore has the complete list already.

Know the versions of the following facts given in class (I might ask for a proof of one or more of these, or a proof of something whose proof is similar), and be able to use them in problems/proofs:

• Theorem 2, p. 21 ("Big Theorem"), and the consequences for the number of elements in the solution set of a linear system.
• The consequence of the preceding, for homogeneous systems (p. 25).
• All the stuff on pp. 38-41 involving matrix arithmetic, transposes, etc., and Theorem 1, p. 47.
• I should have mentioned "dot product" (shaded box, p.44). Know this terminology for the course generally.
• Theorem 2, p. 50.
• Be able and willing to "think big" -- we have seen many situations where this facilitates and simplifies proofs/computations.
• Know the improved version of Thm. 1, p. 58 from class, and the two corollaries of it also given there, and able to use any of these.
• Know the boxed fact on page 59 (after you supply the two periods Nicholson wrongly omitted).
• Know the algorithm on p.61 for determining whether a given square matrix is invertible, and for finding its inverse if it is. Understand why the matrix does what is claimed for it.
• Know the boxed facts on p. 63.
• Know all the boxed facts in section 2.4, and be able to prove some of this stuff and similar stuff, if asked.
Ask in class Friday, DURING class so that everyone can hear the answer, for updates to this test preparation file. I will not answer questions outside class about test content -- that would be giving an advantage to the questioner.

The Chief