I am involved in Proofwiki and I am not mathematician, so I had to learn from scratch what it means to prove things. In one of Numberphile videos Terence Tao said that there are 3 levels of mathematical maturity. On the first level people simply use math language like any other language without paying attention to details, so their arguments quickly become sloppy and ill-defined. On the second level people study maths as a network of definitions and theorems, i.e. people need to understand what are the inputs and outputs of definitions and theorems. This becomes tedious very quickly, because every single logical step you make that is not an identity must be supported by a definition or a theorem. However, after some training people reach the third level where they understand, that some of the arguments can be omitted, so they appear to speak like people on the first level, but now they know what and why they are omitting. I guess you are trying to transition from the first to the second level.
The way I learnt to prove things was to take a book, look at its solutions and try to understand what makes a "proof" a proof. This meant that for each sentence I had to write down which previous theorem allows one to connect arguments in neighboring sentences. Now, it really depends on a book you start with. Some very standard books are actually terrible references for proofs, and finding the right one involves luck. I did not use any actual book about proofs, because no matter how nice a proof for number theory is, if I am working with functional analysis, I need to study proof structure for functional analysis including tiniest lemmas that I could only learn by studying this stuff from scratch.
In your case, I would say that Linear Algebra Done Right is an ok book, but you need to do some extra work. Namely, since you do not yet have a working knowledge needed for proofs, you need to make something like a dictionary, i.e. for a given chapter you need to write out all the definitions and theorems and clearly state what are their inputs and outputs. Once you have this list, given an assumption of an exercise now you can start branching out by connecting all these inputs and outputs. This is the reason why I joined Proofwiki - I use it as a giant dictionary to store such details. To a classically trained mathematician this may sound like overkill - proofs tend to become 2,3 or 4 times longer than what you find in books, but this is what it takes. With time your working knowledge will grow, you will start noticing repetitions in the proofs and you will stop using the same lemma for the n-th time. At this point you will be able to say that you own the second level, at least in some tiny area of maths.
If making the dictionary I presented above is still too complicated, I may provide you some more direct help not constrained by this forum.