So,in Algebra 1, I have to give an example of an exact sequence of homomorphisms in an example of vector spaces and linear maps between them.
I have an ideas. I don't know if any of them work and the main problem at both of them is that I am not able of finding the linear maps. We learned about the connection between these exact sequences and direct sums. The first example:
$0\rightarrow ker(f) \rightarrow V \rightarrow [v]\rightarrow0 $ where V is finite dimensional, f is a linear functional in the dualraum of V and $v \notin kerf $.
This is the first time ever working with exact sequences, therefore I have problems. I have seen that above the arrows there must be functions, I don't know which could be the functions in my example.
I would be really helpful for help. Also If there is any example which might be easier, I would be thankful to know that too in order to get a better idea.
Thanks in advance.
I don't quite understand your sequence, but it looks exact, but only by definition. Like if you have a surjective linear functional $f \colon V \to F$ where $F$ is the field you're working over, then sure you have an exact sequence $$\ker(f) \hookrightarrow V \twoheadrightarrow [v]$$ where you identify $F$ with $[v]$ for some $v \notin \ker(f)$. An important heuristic to keep in mind is that we must have $\dim V = \dim\ker(f) + \dim[v]$ for this to be an exact sequence, so the dimension of the kernel must be one less than $\dim[v]$. As for finding the linear maps, well that depends on what $f$ is. But the cheekiest way I can think to do this would be to choose a basis for $V$ containing $v$ and a basis for $\ker(f)$. So say $\ker(f)$ has basis $\{a,b,\dotsc,z\}$. Then take $V$ to have basis $\{v, a, b, \dotsc, z\}$ then $f$ can be written as
\begin{pmatrix} 1&0&\dotsb&0 \end{pmatrix}
will map nothing to $v$, but be the identity map on the rest of the basis for $\ker(f)$.