Isomorphic and Homomorphic I was wondering if someone could explain the ideas behind isomorphisms and homomorphisms and the difference between them. I understand that a linear map is an isomorphism if it is bijective, and that an isomorphism is a bijective homomorphism but I don't fully understand what a homomorphism actually is.
For example, what does it mean to say that $Hom_{\mathbb{C}}\left(\mathbb{C}^2,\mathbb{C}^3\right)\simeq MAT_{\mathbb{C} }\left(3,2 \right)$ where $MAT_{\mathbb{C}}$ is the $3\times2$ matrix with complex entries. 
After some research, I think that this example is saying that $Hom_{\mathbb{C}}\left(\mathbb{C}^2,\mathbb{C}^3\right)$ is essentially a perfect approximation of $MAT_{\mathbb{C} }\left(3,2 \right)$ however I am struggling to understand what the notation $Hom_{\mathbb{C}}\left(\mathbb{C}^2,\mathbb{C}^3\right)$ actually means. 
Note: I haven't studied a course in group theory and therefore have little knowledge on it, this is for a Linear Algebra Course

A homomorphism is a map between two structures that preserves the operations defined on those structures. For instance, for simplicity, let $(G, \circ)$ and $(H, \diamond)$ be groups. Then a homomorphism is a mapping
$$\phi : G \to H$$
with the additional property that $\phi(g \circ h) = \phi(g) \diamond \phi(h)$ for $g, h \in G$. An isomorphism is a homomorphism which is bijective. This is useful because it means that the two structures are essentially the same. It also means we can move freely between the two structures because a bijective mapping has an inverse. That is, there is a map $\phi^{-1} : H \to G$ that is also a homomorphism. Two structures are isomorphic if there exists an isomorphism taking one to another.
