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);

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

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 V be a vector space of dimension n and let A = {v1, ..., vn} be an ordered basisof V . Suppose w1, ..., wn ∈ V and let (a1j , ..., anj )t denote the coordinates of wjwith respect to A. Put C = [aij ]. Then show that w1, ..., wn is a basis of V if andonly if C is invertible

Matrix A and B are given below:A =1 −3 4 −1 9−2 6 −6 −1 −10−3 9 −6 −6 −33 −9 4 9 0 and B =1 −3 0 5 −70 0 2 −3 80 0 0 0 50 0 0 0 0Assume that the matrix A is row equivalent to B. Find(a) (3 pts) rank A.

Use the fact that matrices A and B are row-equivalent.A = −2 −5 8 0 −17 1 3 −5 1 5−5 −9 13 7 −671 7 −13 5 −3B = 1 0 1 0 1 0 1 −2 0 30 0 0 1 −50 0 0 0 0(a) Find the rank and nullity of A.rank nullity