Any suggestion on how to justify true/false question in linear algebra exams?

Question

I have hard time bringing words on paper when it comes to true false justification of linear algebra problems.

My technique is to use counter example for false and use book theorems for true ones.

Any suggestions?

Answer 1

If the problem is easier in forward path (meaning from premises generate an answer), then follow forward path.

Else if reverse path is easier (meaning from results get to premises) follow that path (as mentioned in comments below, using equivalences and not simple implications to be correct).

Else (try to) find (simple) counter-examples. It is surprising how easy is to find simple counter-examples, in some cases, and effectively prove sth as false (if the positive is given in question).

In any case verify the final result/counter-example you get to be sure. These are just speed hacks for solving (multiple-choice) exam questions

Another set of tips and tricks for speed answering, having to do with actual calculations (and not just inference or equivalences), involves having a "library" of results and simplifications already prepared (in your memory for example) and use them to derive results fast (without having to re-invent every intermediate little theorem or relation) and so on.

Answer 2

This depends on the question itself.

Example 1: True claim

If $W$ is a subspace of $V$ and $\dim(W)=\dim(V)$ then $W=V$

In this type of T/F question which is true you will use what you called book theorems to prove it.

Example 2: True claim

There exist $n>0$ s.t $$ \begin{bmatrix}0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}^{n}=0 $$

where $0$ is stands for the $5\times5$ matrix.

In this case the claim is true, but to prove it we won't use book theorems (probably) but we can calculate that for $n=5$ the equality holds and this will suffice to prove the existence of such $n$

And similarly for false claims - for example if in the first example we were asked if $W\neq V$ then we could just give a counter example, and if we were asked if there exist $n$ such that $I^{n}=0$ then the claim is false but we would have to disprove it and we could not do so with counter examples - we would really have to prove that that there does not exist such an $n$

To make things a little bit more clear:

If a claim is "something will always happen'' - if its true then prove it and if it false give a counterexample

If a claim is "There exist .. such that something happens'' - if its true it suffice to find such a case where it happens and if it false we need to prove that it never happens.

If you have trouble with other types of questions feel free to comment