Math410 (Bayly) Homework 1 (due Friday 1 September)

These are random selections from a lot of similar problems in the text. I recommend you do a lot more if you have the time; the more you see the different situations in which the same kinds of behavior, the better you will understand it.

(1) page 3: 1.1.1 (b, f).

(2) page 9: All parts of 1.2.7. NOTE that some of these are impossible because of the sizes of A, B, C. Here I denotes either the 2x2 or 3x3 identity matrix (square with 1’s on diagonal, 0’ everywhere else).

(3) page 15: 1.3.1(e), 1.3.2(d).

In problems (4), (5), (6) After you find the general solution to the system, identify your particular solution and null vector(s) if any. NOTE there might be null vectors even if there is no solution to the given system! If there are one or more null vectors, which columns of original matrix are basic and which nonbasic? For each nonbasic column, express it as a combination of basic columns.

For 2d systems draw an “aiming vector, target vector” sketch of each system. And if there’s a null vector, sketch the combination of aiming vectors that yields zero.

(4) page 66: 1.8.1(a, b, d).

(5) page 66: 1.8.2 (a, b, c, d). Parts (a,b) do not have solutions as stated; sketch the aiming and target vectors and indicate why there is no solution. Also find the null vector and interpret its meaning in terms of the aiming vectors. You don’t have to sketch for (c); are there any null vectors? In part (d) do the same type of sketch as (a). Find two null vectors and interpret their significance in terms of the aiming vectors.

FINALLY for problem 1.8.2(a,b, c,d), replace the right sides with or and find compatibility conditions on the b’s (so that solutions exist for the linear systems).

ADDED TASKS from 28 SEPTEMBER: For problems involving row reduction:

For the problems that do NOT require row exchanges, identify L and U factors, and verify LU = A.

Also perform row reduction on the transpose of A , check that its rank is the same as rank of A, find its null vector(s) , and check that the solvability condition for really is .