## anonymous 5 years ago Let P₃(R) be the vector space of polynomials over R of degree strictly less than 3. Let T: P₃ (R) →P₃ (R) be defined by T(⨍)= 2⨍’ (a) Show that T is a linear transformation (b) Find a basis for the kernel of T (c) Is T an isomorphism? Justify your answer.

$$T(f+g)=2(f+g)'=2(f'+g')=2f'+2g'=T(f)+T(g)$$ $$T(\alpha f)=2(\alpha f)'=2\alpha f'=\alpha(2f')=\alpha T(f)$$ Hence $$T$$ is linear. $$f\in \ker T\iff T(f)=2f'=0\iff f'=0\iff f\equiv \text{costant}$$ Obviously $$T$$ is not an isomorphims since it is not injective ( because $$\ker T\neq 0$$ )