1. The problem statement, all variables and given/known data Let T:[R[/3]→[R[/3] so that when u=[R][/3] and v=(1,2,1), then T(u)=u×v a) Show that T is a linear transformation. b) Find T((3,0,2)) c) Find a basis for Ker( T ). Give a geometric description of Ker( T ). 2. Relevant equations Properties of a linear transformation: i) T(u+v)= T(u) + T(v) ii) T(ku)= KT(u) 3. The attempt at a solution a) I have proven T is a linear transformation by closing it under both addition and scalar multiplication and am confident in my calculations for this part. b) I am not so confident in this part. Here are my calculations; T(u)= u×v T((3,0,2)) = (3,0,2)×(1,2,1) =i(0⋅1-2⋅2)-j(3⋅1-2⋅1)+k(3⋅2-0⋅1) =i(0-4)-j(3-2)+k(6-0) =(-4, -1, 6) ∴T((3,0,2))=(-4, -1, 6) I had to make an educated guess at how to approach this question as we haven't covered it in our coursework so far. Please let me know if this is correct or correct me if I am on the completely wrong path! c) I don't know how to approach this part. I know how to find a kernel, given a matrix but I don't have a matrix. Should I form a matrix with vectors (3,0,2), (1,2,1) and (-4,-1,6)? I attempted this but found consisted only of the zero vector and subsequently has no basis. Is the because T:[R[/3]→[R[/3] is an orthogonal projection and so the points T maps onto 0=(0,0,0)? If this is the case, should I still include my calculations of the matrix formed by the three vectors or could I just state this information and move on? Any help would be greatly appreciated. I am finding this topic quite difficult to get my head around as I am studying it through correspondence and am very disappointed by the amount of education and support I am getting from the university.