Determine N (T ) and R(T ) for each of the following linear transformations:a) T : P2 → P3, T (f )(x) = xf (x)b) T : P4 → P3, T (f )(x) = f ′(x).

Sure, let's determine the null space (N(T)) and range (R(T)) for each of the given linear transformations.

a) T : P2 → P3, T (f )(x) = xf (x)

The null space N(T) is the set of all vectors in the domain that T maps to the zero vector in the codomain. In this case, the only polynomial in P2 that T maps to the zero polynomial in P3 is the zero polynomial itself. So, N(T) = {0}.

The range R(T) is the set of all possible outputs of T. Since T can take a polynomial of degree at most 2 and map it to a polynomial of degree at most 3, the range of T is all of P3. So, R(T) = P3.

b) T : P4 → P3, T (f )(x) = f ′(x).

The null space N(T) is the set of all vectors in the domain that T maps to the zero vector in the codomain. In this case, any constant polynomial in P4 will be mapped to the zero polynomial in P3. So, N(T) is the set of all constant polynomials in P4.

The range R(T) is the set of all possible outputs of T. Since T maps a polynomial of degree at most 4 to its derivative, which is a polynomial of degree at most 3, the range of T is all of P3. So, R(T) = P3.