(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that S={u,v,w} is a basis of R^{3}.

a) Is {u-v,u+v} linearly independent? Why or why not?

b) Does {u+v,u-v} span R^{3}? Why or why not?

2. Relevant equations

na

3. The attempt at a solution

a) Because we know that u,v,w are linearly independent and span R^{3}, c_{1}u+c_{2}v+c_{3}w =0.

To test u+v, u-v, consider a(u+v)+b(u-v)=0. So, au+av+bu-bv=0, so u(a+b)+v(a-b)=0. Since u and v are linearly independent, a+b=0and a-b=0. So 2b=0, so b=0, so a=0. So u+v, u-v are linearly independent.

b) Because there are only 2 vectors, u and v, the rank can only be 2, so it cannot span R^{3}.

I'm pretty sure I got a) correct, but I'm not sure about b). Thanks!

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# Linear Algebra - independent vectors spanning R^3

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