Ive been on this Linear Algebra problems for so long Please

I\'ve been on this Linear Algebra problems for so long.. Please assist

Determine the values of k for which the set S = {x-l, kx + 1} would span P_1 (polynomials of degree 1 or less), represent linearly independent polynomials.

Solution

a)

P1 has dimension 2 and S has 2 elements. So for S to span P we need that the two vectors in S must be linearly independent

Let,

a(x-1)+b(kx+1)=0

(a+kb)x+b-a=0

Hence, a=b, a=-kb

So there are infinitely many solutions other than:a=b=0 if k=-1

So value k can take are: R\\{-1}

b)

The answer is same as a) as to span we require vectors to be linearly independent polynomials.