Are the transfermations linear? Are they an isomorphism? Why? Solution 1. T(f)=f\' T(f+g)=(f+g)\'=f\'+g\' T(cf)=(cf)\'=cf\' So, T is linear P2 is space of polynomials of degree atmost 2 so T(P2) can have polynomials of degree atmost 1 So it cannot be an isomorphism because P2 has dimension 2 and T(P2) has dimension atmost 1 2. T(f+g)=(f+g)\'+(f+g)\'\'=f\'+f\'\'+g\'+g\'\'=T(f)+T(g) T(cf)=(cf)\'+(cf)\'\'=cf\'+cf\'\'=cT(f) HEnce, T is linear But T is not an isomorphism As T maps a polynomial to a polynomial of degree stricly smaller than itself. HEnce P3 and T(P3) have different dimension so T cannot be an isomorphism..

1. Are the transfermations linear? Are they an isomorphism? Why?
Solution
1. T(f)=f'
T(f+g)=(f+g)'=f'+g'
T(cf)=(cf)'=cf'
So, T is linear
P2 is space of polynomials of degree atmost 2 so T(P2) can have polynomials of degree atmost 1
So it cannot be an isomorphism because P2 has dimension 2 and T(P2) has dimension atmost 1
2.
T(f+g)=(f+g)'+(f+g)''=f'+f''+g'+g''=T(f)+T(g)
T(cf)=(cf)'+(cf)''=cf'+cf''=cT(f)
HEnce, T is linear
But T is not an isomorphism
As T maps a polynomial to a polynomial of degree stricly smaller than itself.
HEnce P3 and T(P3) have different dimension so T cannot be an isomorphism.