Are the transfermations linear Are they an isomorphism WhySo

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.