For each of the following linear transformations T, determine whether T is invertible and justify your answer. а - 26 (). (a) T : C² → C³ defined by T a b+a а — 26) (b) T : R³ → R³ defined by T \c+2b, (c) T : P3(Q) → P2(Q) defined by T(p(x)) = p'(x). (d) T : M2(Q) –→ P3(Q) defined by а b T = ax + ax? + (b+ c)x + d.

- Let ƒ : R² → R be defined by f((x, y)) = 3x + 8y − 1. Is ƒ a linear transformation? a. f((x₁, y₁) + (x₂, y₂)) = f((x₁, y₁)) + f((x2, y₂)) = = + Does ƒ((x₁, y₁) + (x2, y2)) = f((x₁, y₁ )) + ƒ((x2, y2)) for all (x1, Y1 ), (x2, Y2 ) E R² ? choose b. f(c(x, y)) : = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Is f a linear transformation? choose . (Enter x₁ as x1, etc.)

Letf : R² → R be defined by ƒ((x, y)) = −3x – 7y. Is ƒ a linear transformation? a. f((x₁, y₁) + (x₂, y2)) = f((x₁, y₁)) + f((x₂, y2)) = Does f((x₁, y₁) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x₂, y₂) E R²? choose . (Enter x₁ as x1, etc.) b. f(c(x, y)) = c(f((x, y))) = = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Isf a linear transformation? choose

Define the linear transformation T by T(x) = AX. Find ker(T), nullity(T), range(T), and rank(T). O -8 A = 16 17 (a) ker(T) (If there are an infinite number of solutions use t as your parameter.) {[ } (b) nullity(T) (c) range(T) O {(0, t): t is any real number} {(16s, 8t, 17s - 6t): s, t are any real number} O {(s, 0): s is any real number} O R3 O R2 (d) rank(T)