Which of the following are linear transformations? (Choose all that apply) OT:RR defined by T(x) = . OT:R→R defined by T(x) = x+5. OT: R² →R² defined by T(< x, y >) =< 2y, x − y >. OT: Pn →Pn-1 defined by T(p) =p where P is the vector space of polynomials of degree at most n and pi the derivative of p. n

Let P, 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) E P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + ₁x + ao and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) E P3? 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 P3? choose c. Is D a linear transformation? choose (Enter a3 as a3, etc.) + +

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 + ²x² + a₁ + a₁ 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))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆

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.)