c) calculate the matrix representing T with respect to the standard basis e1=(1,0)T, e2=(1,0)T of R2 and the basis {v1,v2,v3,v4,v5} of P4(R). d) find a basis for the kernel of T expressed as linear combinations of the vectors v1,..,v5 and state the dimension of the kernel, ker(T), of T e)find a basis for the image of T, Im(T) and state its dimension
please solve c,d,e
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
a) show that T is a linear m ap
b)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).
c) calculate the matrix representing T with respect to the standard basis e1=(1,0)T, e2=(1,0)T of R2 and the basis {v1,v2,v3,v4,v5} of P4(R).
d) find a basis for the kernel of T expressed as linear combinations of the vectors v1,..,v5 and state the dimension of the kernel, ker(T), of T
e)find a basis for the image of T, Im(T) and state its dimension
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