The basis for $U$ here is $$\{(3,1,0,0,0),(0,0,7,1,0),(0,0,0,0,1)\}$$
This spans the subspace, and it's vectors are linearly independent.
The correct notation here is:
$$\operatorname{Basis }U=\{(3,1,0,0,0),(0,0,7,1,0),(0,0,0,0,1)\}$$
You don't need to post the question and explain what you did in an answer. You should delete this answer and copy what is written here in the question.
@hjhjhj57 But then if I am correct, the answer would be "Yes" which would be silly, and I won't know if I am upvoted for my question or my answer
@MhenniBenghorbal Thank you, does that include my attempt at the notation? Or should it be a list like in the book? $\operatorname{Basis } U = ((3,1,0,0,0),(0,0,7,1,0),(0,0,0,0,1))$ instead of a set?
That's what comments are for. There's also a tag called proof-verification which is exactly for what you're doing. The fact that you're asking means there are some parts you're having trouble with, an answer would have to aim to clarify this parts. So unless you already know the answer, there'd be something to add, as @PVAL just did.
@hjhjhj57 It says that it is acceptable to post your own answers in the help page, and also are you saying I am getting free rep for posting my own answers? I have done this 5 times and I have 21 rep...
Forget what I said about the reputation, I just think is is not the way of asking (and think that users that have been around for a while shouldn't encourage it). And in fact, you're not even sure your answer is an answer, that's what the question is for.
@hjhjhj57 I want to be able to discern if my question or my answer are the upvoted thing and if I get upvoted I know it is the right answer, and if it is downvoted I will delete
Well, I disagree with your way of doing things, but we have a democratic community here, so let's see what happens (more like what meta and the moderators have to say).
In that case I would agree with you, but in the case of that question, I have never seen an answer to such a problem, but felt that it was only one that would work
why not post your attempt of proof in your question (using the proof-verification tag), and wait to see if you get useful feedback. If you only get replies saying "yes" you proceed to answer your own question (there's even a badge for doing this I believe).
don't get me wrong, answering your own questions is actually encouraged by SE. I just think that if you're answering you should be completely positive about your answer.
@hjhjhj57 That would be my second preference, but I feel as though people will put up my answer pretty much and be wasting the time of valuable helpers who could be helping others - I also think I will get a faster reaction for doing it this way
I wouldn't worry about the time of the people. This is a huge community and we're all here because we want to. And about the reaction, well, maybe. I don't know.
I was going through those proof-verification questions, and saw something that hadn't occurred to me: people may even suggest different (and probable better) proofs..