In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as:

$$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$

As far as I can see, this is just the same as:

$$\operatorname{Im} \, (T) := \{ \;Tv \in W : \;v \in V\}$$

Is there any difference in these definitions?

If not, why is the first one used?


Each statement defines the image of the linear tranformation, they just give different ways of describing the SAME EXACT set: the image of $T$.

Each definition uses Set-Builder Notation: notation which allows us to describe any given set of elements in any number of ways, and/or from different perspectives or for different purposes. E.g.:

  • Let $E_1 = \{\; 2k | k \in \mathbb{Z}\;\}$;
    Let $E_2 = \{\;n \in \mathbb{Z} \mid n\equiv 0 \pmod{2}\};$
    Let $E_3 = \{\;n \in \mathbb{Z} : 2\mid n\;\}$.

    Each of $E_1$, $E_2$, and $E_3$ each define the same set of even integers. There is only one set being defined; which definition one chooses depends on context.

Back to your two definitions:

  1. The first definition states "The set of all elements $w \in W$ such that $w$ is the value of $Tv$ where $v$ in some element in the domain $V$."
  2. The second defines the set of all function values $Tv$ that end up in $W$ after $v \in V$ is transformed by $T$".

If both define the same set, as they do, then why use the first?

The first is often used to establish, e.g., surjectivity of a function $f: V\to W$. If $f$ is onto, then for every $w \in W$, there exists a $v\in V$ such that $f(v) = w.$ So it's not uncommon to define the image of a function as it is defined in the first case.


Both definitions are the same as you point out, they just give slightly different ways of thinking about the image.

The first definition views the image as "Things in $W$ reached by the function". The second takes a more constructive approach, it is saying that we can build the image by taking each element of $V$ and put the result after applying $T$ to it into the image.


In certain set-theoretic senses, the first one $$ \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$ is exactly what you get by the Axiom of Separation, while the second one $$ \{ \;Tv \in W : \;v \in V\} $$ is taken as a short-hand way of writing it. So perhaps the author thought the first one would be less confusing to some students.

  • $\begingroup$ The second one can also be construed as an instance of the replacement axiom. $\endgroup$ – Zhen Lin Dec 31 '12 at 4:06
  • 2
    $\begingroup$ Besides, why write the $W$ in the second one. Shouldn't we really write $\{ \;Tv : \;v \in V\}$ instead? $\endgroup$ – GEdgar Dec 31 '12 at 14:49

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