Find all indecomposable projective and indecomposable injective representations of these two quivers over the field $k$ up to isomorphism.
I've drawn my answer in this picture. Can you please check if it is right? I know that there is one more projective and one more injective representation of the first quiver which are mirror images of the first ones.
Best Answer
When you have a injective or projective module supported at a single node, it won't be two copies of $\mathbf{k}$ at that node. It'll be only one copy. So for example your projective representation $0\to \mathbf{k}\sqcup\mathbf{k} \leftarrow 0$ should have only a single copy of $\mathbf{k}$ there.
Also, I'm not quite sure why you're using the $\sqcup$ disjoint union symbol. Usually I see $\oplus$ used instead. But anyways, you can clean up your presentation by writing $\mathbf{k}\sqcup 0$ as just $\mathbf{k}$. Similarly $0 \sqcup 0 = 0$.