# Notation in linear algebra, what are $N(T)$ and $R(T)$

Working through some stuff I found on the web, I came across a notation that I haven't seen in my textbooks.

In this problem, $T: P_4(\mathbb R)\rightarrow \mathbb R^4$ is a linear transformation, and there's a formula given to define it. No problem there.

Then some questions follow, including:

1. Write down a basis for $N(T)$.
2. Write down a basis for $R(T)$.

My question is: What is meant by $N(T)$ and $R(T)$?

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+1. Well posed question! – user21436 Feb 5 '12 at 8:49
Sometimes the null space of $T$ we call the kernel of $T$ too. But if I'm not wrong the null space is a term used mainly in linear algebra whereas the term "kernel" can be used in more settings such as the kernel of a group/ring homomorphism or the kernel of a linear transformation. – user38268 Feb 5 '12 at 9:49

1. $N(T)$ is the null-space of $T$, i.e., $N(T)=\{v:T(v)=0\}$
2. $R(T)$ is the range of $T$, i.e., $R(T)=\{T(u): u\in P_4(\mathbb R)\}$