Finding a Basis of V with Vector v

[stunner5000pt]

find a basis for V that includes the vector v

1 V = R3 , v = (0,1,1)
2 V = P2 , v = x^2 - x + 1
WHere Pn represents all polynomials of degree n

for hte first one
would any three independent vecotrs do?
like maybe (1,0,0), (0,1,1) and (0,1,0)?
is that correct?

for the second one
(1,-1,1) is the vector we have to match
could two other vectors like (1,0,0) and (0,1,1) do?
IS there a systematic method of doing this?

 

[0rthodontist]

Well, both of your answers are correct. I don't know of a systematic method. If you have a span of the vector space you can systematically reduce it to a basis, but you aren't given a span.

Here's a semi-systematic way of doing it: you could take a basis that seems "standard" such as 1, x, x^2 for P2 and then let the basis be the standard normal basis in Rn written with respect to that basis in your vector space, with the exception that the vector you are asked to include takes the place of one of the standard normal vectors, with the proviso that when written as a vector in Rn it does not contain a zero in the place where the standard normal vector it is taking the place of contains a 1.

So for (1, -1, 1) you might use (0, 1, 0) and (0, 0, 1) by that method