Show the image of a linear transformation is in another linear transformation

Question

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.

$T:P_{n-1}(\mathbb{R}) \mapsto P_n(\mathbb{R}),$

$p \mapsto p\cdot(X-\alpha)$

Show T is a linear transformation and the image for T is in V.

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I think I have managed to show T is a linear transformation. But I'm not sure how to proof the image for T is in V. I can sort of argue/see that it is in V but how do I make a proper proof?

Answer 1

1. Linearity

If $p_1,p_2\in P_{n-1}(\mathbf{R})$, then $$ T(p_1+p_2)=(p_1+p_2)(X-\alpha)=p_1(X-\alpha)+p_2(X-\alpha)=T(p_1)+T(p_2), $$ and $$ T(cp)=(cp)(X-\alpha)=c(p(X-\alpha))=cT(p) $$ for any $p\in P_{n-1}(\mathbf{R})$ and any $c\in \mathbf{R}$. You can see also that $T(0)=0$

2. Image of $T$ is in $V$

For any $p\in P_{n-1}(\mathbf{R})$, $T(p)(\alpha)=p\cdot(\alpha -\alpha)=0$, then the image of $T$ is in $V$.