What is the difference between codomain and range?

Question

My book says that if there is a linear transformation $T: V \to V'$, then $V'$ is the codomain of $T$ but it also says that $T[V]$ is the range of $T$. $T[V]$ the same as $V'$?

Answer 1

Consider a linear map $T:\mathbb{R} \to \mathbb{R}$ given by $T(x) = 0$ for all real $x$.

It's clear $T$ is linear. The codomain is indeed $\mathbb{R}$, but the range of $T$ is all points in the co-domain where $T$ maps something, so range of $T$ is $\{0\}$.

Answer 2

The codomain need not be the same as the range. Take any projection operator like $\begin{bmatrix}1&0\\0&0\end{bmatrix}$; its codomain is $\mathbb R^2$ but its range is only the subspace spanned by $(1,0)^T$.

However, it is always true that $T(V)\subseteq V'$ and that the transformation can be restricted to its range ($T': V\to T(V)$) such that range and codomain are equal.

Answer 3

Codomain is a set which the images must belong to.

Range is the set which the images exactly belongs to.

Answer 4

The codomain and range have two different definitions, as you have already stated. The range is the set of values you get by applying each value in the domain to the given Relation.

Range = $\{ T(v)$ for every $v$ in the domain$\}$

The codomain is a set which includes the range, but it can be larger. The range is a subset of the codomain.

Answer 5

https://en.wikipedia.org/wiki/Range_of_a_function

In mathematics, the range of a function may refer to either of two closely related concepts:

• The codomain of the function
• The image of the function

...

As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article. Older books, when they use the word "range", tend to use it to mean what is now called the codomain. More modern books, if they use the word "range" at all, generally use it to mean what is now called the image. To avoid any confusion, a number of modern books don't use the word "range" at all.