httpwwwcalpolyedujborzellCoursesYear201011Fall202010Petersen

http://www.calpoly.edu/~jborzell/Courses/Year%2010-11/Fall%202010/Petersen-Linear_Algebra-Math_306.pdf

Let,u,v be in I

So there is some ,x,y in V so that

f(u)=x,f(v)=y

f(u+v)=f(u)+f(v)=x+y

HEnce, u+v is in I

Let, u be in I and c be a scalar

f(cu)=cf(u)

u is in I so f(u) is in V. V is vector space so cf(u) is in V

Hence, I is a subspace of W

Let, x,y be in K

f(x+y)=f(x)+f(y)=0+0=0

So, x+y is in K

Let, x be in K and c be a scalar

f(cx)=cf(x)=c*0=0

HEnce, cx is in K

Hence, K is a subspace of V