Advanced Calculus fi..

60 Advanced Calculus, Fifth EditionPROBLEMSIn these problems, all vectors are written as column vectors. Also, the following matrices are1. Let the linear mapping T have the matrix A.a) Evaluate T(1.0). T(0, l), T(2, -I), T(-1, 1) and graph.b) Find the kernel of T, determine whether T is one-to-one. and find all x such thatT(x) = (2, 3).c) Find the range of T and determine whether T maps v2 onto v2.2. Let the linear mapping T have the matrix B.a) Evaluate T(I.O), T(0, I), T(1, -I), T(-1, -1)andgraphb) Find the kernel of T, determine whether T is one-to-one, and find all x such thatT(x) = (2,4).C) Find the range of T and determine whether T maps v2 onto v23. Let T map V n into V m and have the matrix F.a) Find n and m.b) Find the kernel of T, and determine whether T is one-to-one.C) Find the range of T and determine whether T maps Vn onto V m4. (a), (b), (c) Proceed as in Problem 3 with matrix G.5. (a), (b), (c) Proceed as in Problem 3 with matrix H.6. (a), (b), (c) Proceed as in Problem 3 with matrix J.7. (a), (b). (c) Proceed as in Problem 3 with matrix K.8. (a), (b), (c) Proceed as in Problem 3 with matrix L9. Let the linear mapping T have the equation y = Cx. For general x, find the angle betweenx and T(x) = y, as vectors in v2, and also compare 1x1 and IT(x)l. From these results,interpret T geometrically. [Hint: Consider x as 3 and y as 3, where 0 is the originof E2 and P and Q are points of E2.]SO. Let the linear mapping T have the equation y = Dx. Regard x as G, y as z, as inProblem 9, and describe geometrically the relation between x and y = T(x).