Could you use the unit-vectors to prove this, or perhaps the dimension of the vector space? dim(V) = n?
This is perhaps an easy question- more of a question about mathematical induction than linear algebra, perhaps:
Let be a vector-space over a field ; are linear independent vectors in . Prove by induction that the vectors , are linear independent.
So, the question is asking to prove that the sum of linearly independent vectors is a vector which is linearly independent.
The condition for linear independence is:
, when and only when for all
So , but how do you make an induction step out of this? For , the proposition is that is also linearly independent:
, so you can bracket the scalars out... hmm? But I still can't see how you can make an induction step. I'm sure it's kind of easy, but that's only making it more frustrating.
the induction is over the number of vectors. for n = 1, there's nothing to prove. suppose the claim is true for n (induction hypothesis) and take any n + 1 linearly independent vectors
suppose if we show that we're done. first note that if then
by induction hypothesis and so we're done. if then we'll have: which is not
possible because are linearly independent. this completes our induction.
So, you're proving the proposition for all of n- per definition- and then you prove it for n+1? But once again, this relies on the definition of linear independence. It's a strange question, I think. But thanks for your help NCA- btw. can you recommend a good text book for a first course in linear algebra?