The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Linearly Independent Set Proof

Date: 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

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   
Associated Topics:
High School Linear Algebra
High School Sets

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.