Let L: V → V be a linear map and let M: V ⊕V → V be a map given by a rule
M((v1, v2)) = L(v1) + L(v2).
Prove that M is linear and that Im(M) is isomorphic to Im(L).
This is my proof so far. Let me know if it needs improving or re-working...
Consider a map K: Im(M) -> Im(L)
We have that M(x1,x2) = L(x1,x2) = L(x1+x2)
Let ζ1, ζ2 be elements of Im(M)
Want to show K(ζ1+ζ2) = k(ζ1) + K(ζ2) for K(ζ1), K(ζ2) in Im(L)
Note: K(cζ1) = cK(ζ1) for all c in the field. So we have that ζ1,ζ2 in Im(M) -> ζ1,ζ2 in Im(L) (since ζ1^-1, ζ2^-1 are in M -> ζ1^-1, ζ2^-1 are in L so ζ1,ζ2 must be in the Im(L))
so K(ζ1),K(ζ2) are in Im(L) -> K(ζ1+ζ2) = K(ζ1) +K(ζ2) (1)
also, we have that cζ1 is in Im(M) -> K(cζ1) is in Im(L) (2)
from (2) we have that K(cζ1) = cK(ζ1)
From (1) and (2) we have that Im(M) is isomorphic to Im(L).