Let L R4 rightarrow R3 be the linear transformation defined

Let L: R_4 rightarrow R_3 be the linear transformation defined by L([u_1 u_2 u_3 u_4]) = [u_1 + u_2 u_3 + u_4 u_1 + u_3]. Is L onto? Justify your answer. Find Ker L. Find a basis for range L. Is L one-to-one? Justify your answer.

Solution

let l: R₄equaltoR₃ be th linear transformation deined by L([u₁ u₂ u₃ u₄])=[u1+u2 u3+u4 u1 +u3]   

Let ~u = u1 u2 u3 and ~v = v1 v2 v3 be any two vectors in R 3 , and c any real number. Then L(~u + ~v) = L u1 + v1 u2 + v2 u3 + v3 : Applying the given rule for L, that must be 4 (u1 + v1) + 4(u2 + v2) (u3 + v3) (u2 + v2) + (u3 + v3) . On the other hand L(~u)+L(~v) = u1 + 4u2 u3 u2 + u3 + v1 + 4v2 v3 v2 + v3 . A quick calculation shows then L(~u + ~v) = L(~u) + L(~v) Now we need to take care of scalar multiplication. L(c~u) = L c u1 c u2 c u3 = c u1 + 4c u2 c u3 c u2 + c u3 = c u1 + 4u2 u3 u2 + u3 = cL(~u), and we are through