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(i) "nontrivial solution" same as "linear dependence" same as "determinant zero" same as "the vectors lie in the same plane".

(ii) "trivial solution" same as "linear independence" same as "determinant nonzero" same as "the vectors do NOT lie in the same plane".

Is the above relations correct? I was doing some exercises in "General Vector Space" and noticed the above. Am I mistaken? If not, do these relations always hold?

Edit: I found this useful post, and it is exactly what I wanted [ Using the Determinant to verify Linear Independence, Span and Basis ]

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Dick Armstrong
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  • It would be helpful if you could clear up some details: What are these vectors, what are they a solution to and what matrix are you taking the determinant of? – preferred_anon Jul 04 '15 at 14:25
  • I am the one that noticed these relations (the book doesn't specify them like that), e.g if a=b, b=c, c=d then a=d, and just like this I found that different exercises had some relation and I just want to make sure that (i) and (ii) is correct. I didn't post any specific problems because I think it would be a very long post and sort of irrelevant, I was hoping I could make it a general post because if the general relations are correct then the above is correct :P – Dick Armstrong Jul 04 '15 at 14:46
  • You should use full sentences. The lists of words you give separated by $=$ signs don't convey a clear message. – Matt Samuel Jul 04 '15 at 15:12
  • Oh, that makes sense, will do! – Dick Armstrong Jul 04 '15 at 15:15

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