Do any of the following define homomorphisms from GL(n, C) to GL(n, C) for n ≥ 2?Give brief explanations.(a) f (A) = AT (b) f (A) = (A−1)T (c) f (A) = A2

Condition of semigroup homomorphism should be ____________ans.

5. Determine whether each of these functions from Z to Z is one-to-one (onto)a) f (n) = n − 1 b) f (n) = n2 + 1 c) f (n) = n3 d) 2nf n     6. Determine whether f : Z × Z → Z is onto ifa) f (m, n) = 2m − n b) f (m, n) = m2 − n2 c) f (m, n) = m + n + 1d) f (m, n) = |m| − |n| e) f (m, n) = m2 − 4 f) f (m, n) = m + n7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 18. Let S = {−1, 0, 2, 4, 7}. Find f (S) ifa) f (x) = 1 b) f (x) = 2x + 1 c) 5xf x     

Let G be a group and f: G→ G be a mapping defined by () = −1 ∀ x ∈ G. Then f is an automorphism if and only if i) G is commutative ii) G is non-commutative iii) G is a finite cyclic group iv) G≠ {e}, e is the identity element of G​

Determine N (T ) and R(T ) for each of the following linear transformations:a) T : P2 → P3, T (f )(x) = xf (x)b) T : P4 → P3, T (f )(x) = f ′(x).

1. Consider the map π1 ∶ FN Ð→ F given byπ1((an)) = a1.(a) Show that π1 is linear