Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Let v1 =1+x+x2 +x3, v2 =1+2x+3x2, v3 =2+6x, v4 =6. Are the vectors v1, v2, v3, v4 linearly independent over each of K and explain why. (i) K=R; (ii) K = F3, the field of integers mod 3; (iii) K = F5, the field of integers mod 5.

Suppose f is the function defined by f(x) = Ax, where a. The domain of f is b. The codomain of f is 24 0 DA. OB. OC. OD. DE. OF. A = 0 1 8 3 -3 0 2 -1 5 For instance, enter R^5 for R5. For instance, enter R^5 for R5. c. Select all of the vectors that are in the kernel of f. You should be able to justify your answers (there may be more than one correct answer DA. OB. OC. D. DE. F. d. Select all of the vectors that are in the image of f. You should be able to justify your answers (there may be more than one correct answer).

Using linear algebra, show that G={p1(x)=1−x+x^2, p2(x)=x, p3(x)=x+x^2} is a basis of P2, the vector space of polynomials of degree at most two with real coefficients.

All the listed vector spaces are constructed over the field of real numbers R and equipped with the usual operations. In which case are the vector spaces isomorphic to each other? The correct answer is (b), or why?