Show that three of the vectors u=(1,1,-1), v=(-1,1,1), w=(1,3,-1), x=(1,1,0) form a basis for space and express the fourth vector as a linear combination of these basis vectors.

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Show that three of the vectors u=(1,1,-1), v=(-1,1,1), w=(1,3,-1), x=(1,1,0) form a basis for space

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In order to form a basis, they must span and be linearly independent.

The problem says show that three of the vectors form a basis. Which three? Here are the first three (u,v,w). Try the vector x in there somewhere and see if you can find a basis.

To show the set spans R^3, we have to show that an arbitrary vector can be expressed as a linear combination



of the vectors in what we can call S. Expressing this equation in terms of components gives us:





Equate componets:







Thus, to show that S spans R^3, we have to show that the system above has a solution for all choices of .

To prove S is linearly independent, we must show that the only solution of



is

So, we have







has only the trivial solution.

We can do this both by seeing if the determinant of the system is not equal to 0.



The determinant is 0, so it does not form a basis.

Try u,v, and x. I think they form a basis.