field regarded as vector space is a cogenerator

Proposition

Let \(K\) be a field.
Then \(K\) is a cogenerator in Category Vect_K, i.e. for two vector spaces \(V,W\) and vectorspace-homomorphisms \(f_1,f_2: V_1 \rightarrow V_2\) with \(f_1 \neq f_2\) there exists a vector space homomorphism \(h: V_2 \rightarrow K\) such that \(h \circ f_1 \neq h \circ f_2\)

Proof

Let \(f_1 \neq f_2\), then there exists an \(v \in V\) such that \(f_1(v) \neq f_2(v)\).
We distinguish two cases

a)

If \(\{f_1(v), f_2(v)\}\) is linearly independent, then we may choose a basis \(\mathcal{B} = \{b_i\}\) containing \(f_1(v), f_2(v)\).
Now define \(h\) on the basis \(\mathcal{B}\) by

\begin{align*} h: V \rightarrow& W \\ b_i \mapsto& \begin{cases} 1 & \mbox{if } b_i = f_1(v) \\ 0 & \mbox{else } \\ \end{cases} \end{align*}

b)

If \(\{f_1(v), f_2(v)\}\) is linearly dependent, then we may choose a basis \(\mathcal{B} = \{b_i\}\) containing \(f_1(v)\).
Now define \(h\) on the basis \(\mathcal{B}\) by

\begin{align*} h: V \rightarrow& W \\ b_i \mapsto& \begin{cases} 1 & \mbox{if } b_i = f_1(v) \\ 0 & \mbox{else } \\ \end{cases} \end{align*}