Serlo: EN: Epimorphisms

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Linear maps preserve linear combinations. We now learn about special linear maps that preserve generators. These are called epimorphisms.

Motivation and derivation

In the article on monomorphisms we considered linear maps which map linearly independent vectors to linearly independent vectors. There we found out that these maps are exactly injective linear maps. Injective linear maps therefore "preserve" linear independence.

Using the linear independence, we could express the intuitive dimension notion in mathematical (linear algebra) terms. There, we also encountered generators. Now: Are there also linear maps that preserve generators?

So let $V,W$ be two $K$-vector spaces over the same field $K$ and $\{v_1,\dots ,v_n\}\subseteq V$ a generator. Now, what properties must a linear map $f:V\to W$ satisfy, in order for $\{f(v_1),\dots ,f(v_n)\}$ being a generator of vector space $W$? For this, we would need to be able to represent any $w\in W$ as a linear combination of $f(v_i)$. That is, we need to find ${\lambda }_1,\dots ,{\lambda }_n\in K$ such that Vorlage:Einrücken Since the map $f$ is linear, this is equivalent to Vorlage:Einrücken So $w$ must be in the image of $f$. This is said to hold for every $w\in W$. Thus $f(V)=W$ is a necessary condition for $f$ to preserve generators.

Is this also a sufficient condition? Let $f(V)=W$. We investigate whether every $w\in W$ can be represented as a linear combination of $f(v_i)$. Because $f(V)=W$ we have for any $w\in W$ a vector $v\in V$ with $f(v)=w$. Since $v_1,\dots ,v_n$ is a generator of $V$, there are some linear combination factors ${\lambda }_1,\dots ,{\lambda }_n$ with Vorlage:Einrücken So we can write $w$ as: Vorlage:Einrücken And hence $w$ is within the generated space of the $f(v_i)$.

Thus, the linear map $f$ preserves generators if and only if $f(V)=W$. Moreover, $f$ satisfies $f(V)=W$ exactly if $f$ is surjective. Thus, a linear map must be surjective to have the generating property. We call surjective linear maps epimorphisms.

Equivalent characterization of epimorphisms

We have already considered in the motivation that surjective linear maps are exactly the maps that preserve generators.
Because the case of finite generators is more important than the general statement, we consider this case first. Then we investigate what we need to change for the general case: Mathe für Nicht-Freaks: Vorlage:Satz

Now we generalize to vector spaces of arbitrary dimension:

We will now be introduced to a second (category-theoretic) characterization of epimorphisms, the possibility of being "right shortened":

Exercises

Mathe für Nicht-Freaks: Vorlage:Aufgabe {{#invoke:Mathe für Nicht-Freaks/Seite|unten}}