et T and S be linear transformations on a finite dimensional vector space V, where S is non-singular. Then,ans.Rank (ST)= Rank(TS) ≠ Rank TRank (ST) ≠ Rank(TS) ≠ Rank TRank (ST) ≠ Rank(TS) = Rank TRank (ST) = Rank(TS) = Rank T Previous Marked for Review Next

Let T be a linear transformation from an n dimensional vector space V to an mdimensional vector space W and let C be the matrix of T with respect to a basis Aof V and B of W . Show that(a) rank(T ) = rank(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)

(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 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

Let 𝑇: 𝑈 ->𝑉 be a linear transformation, then rank 𝑇 + 𝑛𝑢𝑙𝑙𝑖𝑡𝑦 𝑇 =