Thread: Linear Algebra help

is a basis for a three dimensional real vector space V. Show that the set 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 ?

Note that if then

and so

The same can be done to find a linear combination of w's for . Can you finish?

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?

so some how show = ? Still not sure where to start though

Originally Posted by MacstersUndead

Note that if then

and so

The same can be done to find a linear combination of w's for . Can you finish?

But thats not a linear combination ?

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 . If you can show that are linear combinations of w vectors, then you can replace each 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

Combine like terms after you express and 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.