I am doing the following problem for practice to understand linear transformations.
Consider the linear transformation $T:P_2 \rightarrow P_2$ defined by:
$$ T(p(x)) = p(x+1) $$
Where $P_2$ is the vector space of polynomials of at most degree 2. Determine whether T is injective, surjective, or bijective.
The problem I am having trouble on how to start this. Just from using my intuition I would say that the $T$ is bijective partially because it is going from one vector space to the same vector space. But I am unsure of how to show this. If anyone can help me I would greatly appreciate it.
Best Answer
Hint:
Here's a start: to check whether $T$ is injective, assume that $T(p) = T(q)$. What's that tell you about $p$ and $q$ an their relationship to each other? If this, through your knowledge of algebra, implies that $p = q$, then $T$ is injective.
[This technique is really quite general: it's what you should first do whenever you try to prove something's injective, unless you know quite a lot more (e.g., you know it's a matrix transformation with full rank, ...)]