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.

Question

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.

watchmath · 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 $\ker T\neq 0$ )