1. Consider the vector space of polynomials 1+x^3 , 1-x+x^2, 2x, 1+x^2 Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors? 3. I have tried to solve this by letting a1 = 1+x^3 a2 = 1-x+x^2 a3 = 2x a4 = 1+x^2 Then I let (alpha)a1 + (beta)a2 + (gamma)a3 + (delta)a4 = 0 (alpha)(1+x^3) + (beta)(1-x+x^2) + (gamma)(2x) +(delta)(1+x^2) = 0 (alpha +beta+delta) + x(2gamma - beta) + x^2(beta + delta) + x^3(alpha) = 0 So alpha + beta+ delta = 0 2gamma - beta = 0 beta + delta = 0 alpha = 0 But I can't solve for beta delta and gamma so how do I know if their independent or dependent?