Math410 (Bayly) Homework 1 (due Wednesday 14 July)

If you’re not used to systematically solving systems of linear simultaneous equations, you can refresh your memory by reading section 1.1 and doing parts (a)-(g) of problem 1.1.1.

I did not yet talk about “augmented matrices” or “row-echelon form”, which are basically ways of streamlining the elimination process. If you know it, use it; if not, just solve the systems by straightforward algebra (unstreamlined).

I also mistakenly assigned 1.8.8 on ranks of matrices. You don’t (necessarily) know that yet! Don’t do this problem yet.

So here is what is due for Wednesday 14 July:

Section 1.8, problems 1.8.1(a), 1.8.2(a). After you find the general solution to the system, identify your particular solution and null vector(s) if any. Draw an “aiming vector, target vector” sketch of each system. If there is a null vector, sketch the combination of aiming vectors that yields zero.

Section 1.8, problem 1.8.3 (a,c,d). Part (a) does not have a solution 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 1.8.3(a,c,d), replace the right sides with or and find compatibility conditions on the b’s (so that solutions exist for the linear systems). Find the general solutions in each case, and say how many solutions there are.