Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P3 → P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + α₁x + αº and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = î Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ◆ c. Is D a linear transformation? choose ◆

The points P (x1, y1), Q(x2,y2) and R (x,y) are collinear. Use the gradient formula to show that (x-x1) (y-y2) =(y-y1)(x-x2).

Let P = (1,3) and Q = (2,6) be points in the plane. Find a vector-valued function R(t) = R0+tv such that R(t) describes the line through P and Q.

Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)