Module over a path algebra Let $k$ be a finite field and let $A$ be the algebra of upper triangle $2$ by $2$ matrices.
$$A = \left\{ \begin{bmatrix}
a & b \\
0 & d 
\end{bmatrix} | \  a,b,d \in k \right\}$$
(a) Show a left module $M$ is determined by two vector spaces $V$, $W$ and a linear map $T: V \longrightarrow W$, so we can write $M = T : V \longrightarrow W$
My idea: As $A$ is a path algebra $2 \xrightarrow{\alpha} 1$, I can choose $W= e_{1}M$, and $V=e_{2}M$, and mapping $T_{\alpha} = V \longrightarrow W$, which is multiplication to $\alpha$.
I don't know if this is determining my module. Can you help me with that? Also, I want to deduce the matrix form related to my $T_{\alpha}$, but I don't know how I should do that.
(b) Describe a homomorphism of two left $A$-modules given by data such as above.
My idea: If $f : M \longrightarrow M^{'}$, then $f = f_{1} \oplus f_{2} : M_{1} \oplus M_{2} \longrightarrow M^{'}_{1} \oplus M^{'}_{2}$
It was somehow natural to define that but I am not sure it is right.
(c) Suppose $M$ is finite-dimensional. Show that we can change the basis in both $V \simeq k^n$ and $W \simeq k^{m}$ to write $T$ as
$$ T = \begin{bmatrix}
I_{d} & 0^{d \times (n-d)} \\
0^{(m-d) \times d} & 0^{(m-d) \times (n-d)}
\end{bmatrix}$$
Honestly, I have no idea what is happening here. How my module could be not finite-dimensional at first place, which now is changed to be finite. My guess is when $M$ has a basis, we need to just talk about them, which my $T_{\alpha}$, will affect on the basis. As the number of the basis is $d$, so we expect $I_d$ and other entries $0$, but I think that the $T$ here, has a different definition from mine!
(d) Conclude that ant finite-dimensional left $A$-module is direct sum of three types of modules called $S_0$, $S_1$, and $P_1$ of dimension $1$, $1$, and $2$.
My idea: As I have read about Quivers and path algebras, I deduced that there are two modules here, $M_{1}$ with dimension $2$, and $M_{2}$ with dimension $1$, which their direct sum is $M$. But I couldn't figure out the third one! It seems there is something that I am missing in general.
(e) Which of $S_0$, $S_1$, and $P_1$ are simple or cyclic?
My idea: As $M_{1}$ has two bases I think that it can't be simple, since, with each base, we can produce a submodule. $M_{2}$ has one basis, so it is cyclic. About the third one, I don't know what it is!
(f) Describe the free $A$-module $A$ as direct sum of $S_0$, $S_1$, and $P_1$.
My idea: Maybe it is just saying that $A$ itself is just all composition of paths length $0$ and $1$. But still, I am missing the third one!
It will be great if you help me to get a complete understanding of each part. I spent a lot of time working on this question but I still feel I am confused and missing a lot of details.
Thank you so much.
 A: I answer only the first four.
(a) I assume you used $e_1=\pmatrix{1&0\\0&0}$ and $e_2=\pmatrix{0&0\\0&1}$.
Then call, say, $f:=\pmatrix{0&1\\0&0}$, so that $e_1,f,e_2$ is a basis of $A$ over $k$.
$T$ is then defined by $v\mapsto fv$. 
Note that $e_1f=f$, so $fv\in e_1M=W$.
Also, since $e_1+e_2=1$, we have $m=e_1m+e_2m$, consequently, as $k$-vector space, we have $M\cong W\oplus V$.
For the converse, assume $V,W$ and $T:V\to W$ are given, and define an $A$-module structure on $W\oplus V$ by
$$e_1v=e_2w=fw=0,\quad e_1w=w,\ e_2v=v,\quad fv=Tv\,.$$
Alternatively,
$$\pmatrix{a&b\\0&c}\cdot\pmatrix{w\\v}:=\pmatrix{aw+b\,Tv\\cv}\,.$$
(b) A homomorphism between $A$-modules given by data $V,W,T$ should be a commutative diagram of linear maps
$$\matrix{ V& \overset T\to& W\\
\downarrow&&\downarrow\\
V'&\underset{T'}\to&W'}$$
(c) The above construction works for infinite dimensional $V$ and $W$ as well, so it makes sense to restrict to finite dimensions.
Given a linear map $T:V\to W$, first choose a basis $(k_i)$ of its kernel, extend it to a basis of $V$ by elements $v_j$, then show that $(Tv_j)$ is a basis of the range of $T$, finally extend it to a basis of $W$ by elements $w_s$.
(d) For any fixed indices $i,j,s$, consider the following submodules
$$S_0:=\langle k_i\rangle\\
S_1:=\langle w_s\rangle\\
P_1:=\langle v_j\rangle\,.$$
Describe their $A$-module structure by the actions by $e_1,f,e_2$ on the generator element.
