You just need to do a bunch of proofs most likely. Usually, for elementary proofs, you should write down everything that's given (everything you "know") and translate any definitions into their full meaning and then play around with things.
There are some general tips for proofs, such as:
If the proposition involves a negative, such as "there does not exist" or something along those lines, you should give contradiction a try
If the proposition is of the form: if p then (q or r), you assume not q and try to prove r, (or assume not r and try to prove q)
If the proposition states something like "there exists..." you're probably going to have to build/provide it to end the proof
If you go on to take a "proofs" class this will become much clearer and easier. It is pretty challenging to try to learn the linear algebra and the proofs behind it at the same time because linear algebra courses dont normally focus of the method of proving things, like a proofs class would