6. Show that there does not exist a lincar transformation T: R5 R5 such that R(T) N(T). 7. Let A be a 5 3 matrix having the null space spanned by the column matrices Find the rank of A. , and 3 8. A linear transformation T: R2 R3 is defined by ===== T (2) - B = x1 x2 2x1 21 Find the matrix A such that T(X) = AX, where X = Describe the null-space of X2 T. 9. Let V be a finite dimensional vector space and let T be a linear transformation from V to itself. Suppose that rank (T2)=rank(T). Prove that the range and null space of T are disjoint, i.e., have only the zero vector in common. 10. Let V and W be vector spaces over the field F and T: VW be a linear transformation. If {u, u2, un} are in V such that {T(u), T(u2),..., T(un)} are linearly independent in W, then show that {u, u2,...un} are linearly independent in V. Does the converse hold true? Justify.