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February 19th, 2010, 10:57 AM   #1
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Linearly dependent Theorem

I think I have the wrong interpretation of the following theorem: The nonzero vectors v1, v2,...,vn in a vector space V are linearly dependent if and only if one of the vectors vj (j ? 2) is a linear cominatin of the proceeding vects v1, v2,..., vj-1. My interpretation in summary: a group of vectors is linearly dependent when there is a vector vj (being the second or greater vector) in a group of vectors is a combination of the previous vectors. There is a vector vj (being the second or greater vector) in a group of vectors that is a combination of the previous vectors when the group is linearly dependent.
Now why does vj (j ? 2)? I would really appreciate if anybody could clear up my confusion and correct me of any errors. Thanks
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February 19th, 2010, 02:47 PM   #2
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Re: Linearly dependent Theorem

Quote:
Originally Posted by tmlfan_179027
I think I have the wrong interpretation of the following theorem: The nonzero vectors v1, v2,...,vn in a vector space V are linearly dependent if and only if one of the vectors vj (j ? 2) is a linear cominatin of the proceeding vects v1, v2,..., vj-1. My interpretation in summary: a group of vectors is linearly dependent when there is a vector vj (being the second or greater vector) in a group of vectors is a combination of the previous vectors. There is a vector vj (being the second or greater vector) in a group of vectors that is a combination of the previous vectors when the group is linearly dependent.
Now why does vj (j ? 2)? I would really appreciate if anybody could clear up my confusion and correct me of any errors. Thanks
The definition you gave is a funny way of stating it. Usual definition has any vector being a linear combination of others. However, given such a combination, you can rearrange the vectors to have the vector with the highest index equal to a linear combination of others.
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February 19th, 2010, 03:18 PM   #3
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Re: Linearly dependent Theorem

Thank you. This clears a lot up. I really made it more complicated then it needed to be.
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