Working with linear operators

a) If you multiply the matrix ({a,b},{c,d}) by n, you get T({na,nb},{nc,nd}) = na + nd = n(a+d) = nT({a,b},{c,d}). That's one property of linearity. The other is that T(({a1,b1},{c1,d1}) + ({a2,b2},{c2,d2})) = T({a1,b1},{c1,d1}) + T({a2,b2},{c2,d2}). You can check this yourself.

b) The kernel of T is the set of matrices M for which T(M) = 0. That means all matrices with the one restriction that a + d = 0. You can show that kerT has 3 dimensions (4 dimensions for the general space of 2x2 matrices minus 1 dimension because there is one linear relation between the variables.) You can see that a, b, and c can vary independently, but d = -a. So a suitable basis would be B = { ({1,0},{0,-1}), ({0,1},{0,0}), ({0,0},{1,0}) because the matrices that satisfy T(M) = 0 are those that can be written in form a({1,0},{0,-1}) + b({0,1},{0,0}) + c({0,0},{1,0}).

c) One complement of kerT is the set of vectors which are orthogonal to every vector in kerT. It has dimension 4 - 3 = 1. You can show that ({1,0},{0,1}) is orthogonal to all three vectors in the basis B defined above. So the set C = { ({1,0},{0,1}) } is a suitable basis for complement(kerT).

d) By "putting together" they mean putting all four vectors into one set, the union of B and C. Note that since we have a 4-dimensional vector space, you will need a 4x4 matrix to describe T.

I hope that is easier to understand than your lecturer. :-)