The question is (the row vectors I show are actually written as column vectors on the homework):

Consider the real vector space R>3. One basis we can use is called the Standard Basis. This is the basis

B={|e1>=(1 0 0), |e2>=(0 1 0), |e3>=(0 0 1)}

Show that the set of vectors

B'={|a1>=(1 0 -1), |a2>=(1 2 1), |a3>=(0 -3 2)}

is also a basis for R>3. Express each of the vectors of the standard basis in terms of these new basis vectors. (Hint: To check if you are on the right track, you should get that

|e1>=-7/10|a1> + 3/10|a2> + 2/5|a3> = (-7/10 3/10 2/5)

but you need to show this and find the other two.

I need a little help getting started. I tried: B = {i+j,i-j,k} where i+j=|v> and i-j=|w> and i=1/2(v+w) and j=1/2(v-w). I have also tried combing a1, a2 & a3 to be a 3x3 matrix and multiply by each e, but that didn't work either.

Thanks!