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## Homework Statement

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?

## Homework Equations

na

## 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=

**0**and 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!