Determine whether the function involving the n × n matrix A is a linear transformation.T: Mn,n → Mn,n, T(A) = A−1

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).

(I) The function T : M22→R defined by T(A) = det A is a linear transformation.(II) The composition of two linear transformations is also a lineartransformation.(III)T here is exactly one linear transformation T : V →W for which T(u + v) = T(u − v) for all vectors u and v in V.(IV) The inverse of a one-to-one linear transformation is a lineartransformation.A. II, III. IV True and I FalseB. II, III True and I , IV FalseC. I, III, IV True and II FalseD. II, . IV True and I, III False

Let T𝑇 and S𝑆 be linear transformations defined by T(x,y)=(2x−3y,−y)𝑇(𝑥,𝑦)=(2𝑥−3𝑦,−𝑦) and S(x,y)=(x−y,x)𝑆(𝑥,𝑦)=(𝑥−𝑦,𝑥). What is the matrix corresponding to the linear transformation T∘S𝑇∘𝑆?

Let C be an m × n matrix and let T : Rn → Rm be the linear transformation definedby C. Show that the matrix of T with respect to the standard bases of Rn and Rmis C

Define the linear transformation T by T(x) = Ax. Find ker(T), nullity(T), range(T), and rank(T).A = 4 −2 1 11 −1(a)    ker(T) (b)    nullity(T)(c)    range(T){(s, t, s − 3t): s, t are any real number}R3    {(s, t, 0): s, t are any real number}{(3s, 3t, s − t): s, t are any real number}R2(d)    rank(T)