4.1.1

To find the kernel of $$T$$ simply solve $$T(x) = 0$$, a basis for this is e.g. $$((-14, 10, 14, 6, 0)^t, (0,-5,-1,0,2)^t).$$ We know that the columns of the matrix spans the image, simply remove columns until we get an linearly independent set, e.g. $$((1,2,1,0)^t, (2,0,1,3)^t, (-1,0,3,2)^t).$$ We know $$T: \mathbb R^5 \to \mathbb R^4$$, so in summary dim(dom($$T$$)) = 5, dim(ker($$T$$)) = 2, and dim(im($$T$$)) = 3. So 1.7 holds.