B1. (i) Let T : V ? W be a linear transformation between finite dimensional vector spaces over a field F (a) State the definition of the kernel of T (b) State the definition of the image of T. (c) State the dimension theorem for T, which relates the dimension of its kernel to the dimension of its image. (d) Assume that kerT-{0} and?1, zn ls a linearly independent sequence in V Prove that T(,T(n) is a linearly independent sequence in W. (ii) T : M2(R) ? M2(R) be the linear transformiation defined by T(A) AT, where AT denotes the transpose of A. (a) Prove that T is bijective. (b) Find a basis of M2(R) of eigenvectors for T. (c) Compute Tls, the matrix of T with respect to the basis found in part (b). (d) Find the coordinates of4 with respect to the basis found in part (b)
