Let n>1 be a natural number and let $\alpha \in \mathbb{R}$ be a real scalar. Let V be a subset and subspace of the vector space $P_n(\mathbb{R})$. Define V as
$$ V=\{p\in P_n(\mathbb{R}):p(\alpha)=0\} $$ Let T be function linear transformation.
$$ T:P_{n-1}(\mathbb{R}) ↦ P_n(\mathbb{R}),$$
$$p↦p*(X−α)$$
Show that the Ker(T) is the zero vector space and show that T induce a isomorphism between $P_{n-1}(\mathbb{R})$ and $V$
To show that the Ker(T) is the zero vector space, I know that the I have to find the p, such that $T(p) = 0$. For the kernel would it be $Ker(T)=\{p \in P_{n-1}(\mathbb{R})|T(p)=0\}$ this means $0=p(X-\alpha)=\{p|p*(X-\alpha)=0\} $ so $T(p) = 0$ when $X = \alpha $ right?
I am fairly uncertain how to prove that T induce a isomophism between $P_{n-1}(\mathbb{R})$ and $V$. Hope somebody can verify or disprove the argument for the Ker(T) and help with the isomorphism part.