Here's a long one
Consider the mapping T : R2 à R3 defined by T(x,y) = (x+y , x+2y , 2x+y). Moreover, let S : R3 à R5 defined by S(x,y,z) = (x , 0 , y , z , x+y+z). Answer the following.
(A.) Prove that T is linear.
(B.) If a is the standard basis for R2 and b is the standard basis for R3, then find [T] ab.
(C.) Now let g be the standard basis for R5. Compute [S]bg.
(D.) Compute [ST]ag, where ST := S of T. Do this computation in two ways: once by the way of matrix multiplication and then finding the associated matrix directly.
(E.) Use the result of part (D.) to find ST(1,2).
(F.) Find the kernel of both T and S and compute the associated nullities. Similarly, find the image of both operators and compute the associated ranks.