Determine whether the set $\{v_1 + v_2 - v_3, 2v_1 + 2v_3, -v_1 + v_2 - 3v_3\}$ is linearly dependent or independent. We had a question on our last test that was very similar to this and I only got $2$ points of $6$ and I want to make sure I do it right this time. Here's my solution to that one: 
Let $v_1, v_2,$ and $v_3$ be three linearly independent vectors. 
My teacher told me that to qualify for full credit every detail of my suloution must be presented and the logical steps that lead to my conclusion must be clear, justified, and readable. 
Here's what my solution would be to the question, but i don't think it's enough. 
Here's my updated solution, this should be enough right? 
Thanks guys
 A: i think you can explain what you are doing in words. you have not made clear that you understand what linear dependence of vectors mean. true, you are showing some row reduction, but why and how does it relate to the question.
one way to do is to explain how you determine linear independence by supposing $$a(v_1+v_2-v_3) + b(2v_1+2v_3) + c(-v_1 + v_2 - 3v_3) = 0 $$ rewrite as $$v_1(a+2b-c)+v_2(a+2c)+v_3(-a+2b-3c)= 0 $$ now, use the linear independence of $v_1, v_2, v_3$ to get three equations. 
do you see the point?
A: If one understands why you came up with the original matrix and how the linear dependence follows from your fast reasoning and writings derived from that matrix, then it's fine. However, to make a proof that is unquestionable, you would probably need more detail. However, if it makes you feel any better, if I were a professor I would be inclined to give you credit for correct intentionality based on what you wrote and I would give you full credit. Time is short on tests, after all.
A: You can find if a set is linearly dependent or independent by calculating the determinant.
If it's zero, then the set is dependent. 
If it's non-zero, then the set is independent.
