# How do I prove linear transformation from polynomial to real number vector spaces?

I'm really confused about this homework problem for my linear algebra class, we never really went over this so I have no idea how to answer it. Please try to give a simple explanation since I'm just in the first few chapters of my linear algebra course. Thanks!

Let $\mathbb{P}$ denote the space of all polynomial functions and let $T \colon \mathbb{P} \to \mathbb{R}$ be defined by $$T(p) = \int_a^b p(x) \,\text{d}x.$$ Prove that $T$ is a linear transformation from $\mathbb{P}$, the vector space of polynomial functions, into $\mathbb{R}$, the vector space of real numbers.

• Do you know that a linear transformation from one vector space to another vector space, is? – John Smith Sep 30 '16 at 17:01
• For showing that $T$ is linear you need to show that $T(p+q)=T(p)+T(q)$ and $T(cp)=cT(p)$ for $c \in \mathbb R$. Have you tried to show these conditions?If yes,where are you facing problems? – Math Lover Sep 30 '16 at 17:18
• @JohnSmith Yes, roughly, though the book (linear algebra and its applications 4th ed) never mentions anything about polynomial vector spaces, which makes this really confusing. – Squirrelstilts Sep 30 '16 at 17:22
• @VictorBarg I did try to show those conditions, but I thought I might have to do something else with the vector spaces. I don't really know though. – Squirrelstilts Sep 30 '16 at 17:24
• @Squirrelstilts: you don't need to do anything else,you just need to show the conditions which i wrote in my last comment. – Math Lover Sep 30 '16 at 17:26