# Linear Algebra help

• Mar 3rd 2013, 11:00 AM
Tweety
Linear Algebra help
$\displaystyle {v_{1}, v_{2}, v_{3} }$ is a basis for a three dimensional real vector space V. Show that the set $\displaystyle {v_{1} + v_{2} , v_{2} + v_{3}, v_{3}+ v_{1} }$ is also a basis for V.

I am finding it really hard ot understand the concept of finding bases, and my notes dont make sense, am not even sure where to start, do i first find a spanning set ?
• Mar 3rd 2013, 11:16 AM
Re: Linear Algebra help
Note that if $\displaystyle w_1 = v_1 + v_2, w_2 = v_2+v_3, w_3 = v_3+v_1$ then

$\displaystyle w_1-w_2+w_3 = 2v_1$ and so $\displaystyle v_1 = (1/2)(w_1-w_2+w_3)$

The same can be done to find a linear combination of w's for $\displaystyle v_2,v_3$. Can you finish?
• Mar 3rd 2013, 12:39 PM
ILikeSerena
Re: Linear Algebra help
Hi Tweety! :)

To show the new set is a basis for V, you should show that each of the original vectors can be constructed from the new set.
Does that make sense?
• Mar 6th 2013, 07:51 AM
Tweety
Re: Linear Algebra help
so some how show $\displaystyle v_{1}$ = $\displaystyle v_{1} + v_{2}$ ? Still not sure where to start though
• Mar 6th 2013, 07:52 AM
Tweety
Re: Linear Algebra help
Quote:

Note that if $\displaystyle w_1 = v_1 + v_2, w_2 = v_2+v_3, w_3 = v_3+v_1$ then

$\displaystyle w_1-w_2+w_3 = 2v_1$ and so $\displaystyle v_1 = (1/2)(w_1-w_2+w_3)$

The same can be done to find a linear combination of w's for $\displaystyle v_2,v_3$. Can you finish?

But thats not a linear combination ?
• Mar 6th 2013, 08:39 PM
The new set that ILikeSerena is referring to is your new set of w vectors. If you can show that each v vector is a linear combination of w vectors, then you are done. You are done because let t be an arbitrary vector in V. since t is a vector in V, with v's as the basis, t can be expressed as $\displaystyle t = av_1 + bv_2 + cv_3$. If you can show that $\displaystyle v_1,v_2,v_3$ are linear combinations of w vectors, then you can replace each $\displaystyle v_i$ with w's.
I have given you one such combination. (Verify by replacing each w vector by the sum of v vectors and algebra) In this case by simple replacement $\displaystyle t = a[(\frac{1}{2})(w_1-w_2+w_3)] + bv_2 + cv_3$
Combine like terms after you express $\displaystyle v_2$ and $\displaystyle v_3$ in terms of w's and observe the arbitrary vector t in the vector space V is a linear combination of w's, hence the set of w vectors is a basis.