Let f be a liver function with domain V and codomain V Let K

Let f be a liver function with domain V and codomain V. Let K be the get of all vectors is V such that f (v) = 3 V prove that K is a subspace of V.

Solution

1.

0 belongs to K because since F is a linear map. Hence, F(0)=0

2.

Let, u and v be in K

So,

F(u+v)=F(u)+F(v)       (Because F is linear)

F(u+v)=F(u)+F(v)=3u+3v=3(u+v)

Hence, u+v is also in K

3.

Let c be a real number and u in K

F(cu)=cF(u) , because F is linear

F(cu)=cF(u)=3cu

Hence, cu is also in K

Hence, K is a subspace of V