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January 10th, 2010, 05:12 AM   #1
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linear combinations

I'm doing a little bit of self guided study in linear algebra. This question isn't anything too complex as I'm in the first chapter of an old text book I picked up. It mentioned the concept of linear combinations of vectors with two components in them like [ 5, 6 ] and gave an example of linear combinations of two vectors that aren't parallel. An example is -2[3,1] + 4[1,2] = [-2,6] .

They don't mention it(in the first chapter anyhow) but it seems to me that if you let the coefficients be any real number you could make any/all the vectors that have two components from linear combinations of two nonparallel vectors with two components. I'd also imagine that it would extend to vectors with 3 components, for example you would need three vectors that aren't parallel to build any vector with 3 components.

If this is true, the proof must be in the book somewhere, but my impatience is getting the best of me. Do any of you know if what is described above is true?
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January 10th, 2010, 02:16 PM   #2
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Re: linear combinations

In general, the requirement is that the vectors be linearly independent. For 2-d this is equivalent to not being parallel.
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