Expert: Scott A Wilson - 11/6/2011

Question
Hi, I'm really stuck on a question for homework (I'm a first year undergrad) on vector spaces. I don't really understand the concepts behind the question, I've been struggling a lot with lineard independence etc. The question asks,
Suppose that q0,q1,q2 are polynomials in R2[x] such that qj(2)=0 for all j=0,1,2. Prove that (q0,q1,q2) is not linearly independent in r2[x]. Hint: suppose for the sake of contradiction they are independent, what does this imply for an arbitrary polynomial p element of R2[x]? Also, use the fact that if V is finite dimensional every linearly independent list of vectors in V with length dimV is a basis of V.
I don't get what I'm supposed to do, any help with this would be hugely appreciated.
Adam

Answer
To be linearly independent, all three vectors need to form a 3x3 invertible matrix
in which the rows are linearly independet.

I take it the first two vectors { q(0), q(1) } or non-zerp amd
the 3rd one { q(2) } is all zeros.

This matrix is not invertible since it has one row that is all 0's.

Another way to look at it might be to say q0 = (a,b,c) and q1 = (d,e,f), with q2 = (0,0,0).
Clearly, if we take v = (a+d, b+e, c+f+1), it can't be formed by any linear combination.
In other words, there are no factors g, h, k such that v = g*q0 + h+q1 + k*q2.