Linearly Independent Set ProofDate: 01/24/2001 at 19:37:24 From: sanaz golban Subject: Prove set is linearly independent I have not been able to figure this one out. Assume that in a vector space V, the vectors u and v are linearly independent. Prove that the set {2u-v, u+5v} is linearly independent. I have a hard time with proofs as well, so if you have any tips or steps that will help me solve any proofs, please let me know. Thank you. Date: 01/24/2001 at 19:48:12 From: Doctor Schwa Subject: Re: Prove set is linearly independent To prove something is linearly independent, you want to prove that if a * (first vector) + b * (second vector) = 0, then a and b are both zero. Similarly by definition of linearly independent, you know that if xu + yv = 0, then x = y = 0. So, suppose a(2u - v) + b(u + 5v) = 0. By distributing, you can find that some combination of u and v is equal to zero, so you know that the coefficients are zero; that is, 2a + b = 0 and -a + 5b = 0 also, from which you can prove that a = b = 0. What method did I use for this proof? I saw some complicated words, "linearly independent." Not only that, but those words were used twice, so I wrote down the definition, and wrote down what the "if-then" part of the definition said, and then found that I was already almost done with the proof. Most proofs in linear algebra work pretty quickly that way; look up the definition of the words in the proof, figure out what the "if" part and the "then" part are (that is, what are the givens? what is the goal?) and you're usually at least halfway done with the proof. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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