let T: P4(R)→R2 be the map given by T(f)=(f(1),f"'(1)) where P4(R) is the set of polynomials of degree 4 or less and f"'(1) is the evaluation pf the third derivative of f at 1, such as, d3f/dx3|x=1 1) show that T is a linear m ap 2)show that the set {v1,v2,v3,v4,v5}={1+x,1-x,x2+x3,1-x4,x2+x4} is a basis for P4(R).

let T: P₄(R)→R² be the map given by T(f)=(f(1),f"'(1)) where P₄(R) is the set of polynomials of degree 4 or less and f"'(1) is the evaluation pf the third derivative of f at 1, such as, d³f/dx³|_x=1

1) show that T is a linear m ap

2)show that the set {v₁,v_2,v₃,v_4,v₅}={1+x,1-x,x²+x³,1-x⁴,x²+x⁴} is a basis for P₄(R).