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2 $begingroup$ I have serious doubts - I will be very grateful if someone will help me here Investigate whether the given transformation is a monomorphism / epimorphism. Find its image and kernel. $$ F in L(mathbb R[x]_3,mathbb R[x]_3), F(p)(t) = p(t+1) - p(t) $$ My try Okay, let $$ p(t) = ax^3 + bx^2 + cx + d $$ then $$F(p)(t) = ... = 3ax^2 + 3ax + 2bx + 3 = x^2(3a)+x(3a+2b) +3 = a(3x^2+3x) + b(2x) + 3 $$ Is it monomorphism? let $m_0 = 0 wedge m_1 = 3a wedge m_2 = 3a+b wedge m_3 = 3$ so $a = frac{m_1}{3} wedge b = frac{m_2-m_1}{2} $ so $a$ and $b$ are determined unambiguously so it is monomorphism It is not epimorphism because $ x^3 inmathbb R[x]_3 $ but I can't get $x^3$ in use of $F$ $ker F = left{ 0 right} $ because of part $ 3 $ it will be never polynomial zero If it comes