Representation theory of Lie algebras A representation of the Lie algebra $\mathfrak{g}$ on the vector space $V\ne \{0\}$ is a Lie algebra homomorphism $$\rho: \mathfrak{g} \to \mathfrak{gl}(V),\quad x\mapsto \rho(x)$$

So is the representation actually $\rho$? Where since this is mapping into the general linerar Lie group $\rho$ is a matrix? I am having trouble understanding what a representation actually is. So we have a map from a Lie algebra to a Lie algebra that is a homomorphism, and that homomorphism is the representation and is implicitly a matrix, is that right?

The idea is that we convert mathematical objects and their operations to matrices so we can apply linear algebra it seems - so I suppose the homomorphism is just taking us to matrices, not necessariy being a matrix? I guess this is part of my confusion.
 A: Those formulas just tell you that $\rho$ is a function that turns elements into matrices; that's what a representation is, a recipe for turning elements into matrices.
The set $\rho(\mathfrak{g})$ is a set of matrices that are "structurally similar" to your algebra $\mathfrak{g}$, and it's tempting to blur the distinction between the representation (the recipe) $\rho$ and its end result, the set of matrices $\rho(\mathfrak{g})$. While closely related, they're not exactly the same thing. For example, there may be different recipes to turn the same algebra into the same set of matrices.
It's somewhat similar to how we sometimes blur the distinction between a curve -- say, a function $\gamma : [a,b] \to \Bbb R^2$ -- and its image $\gamma([a,b]) \subset \Bbb R^2$. Technically the two are different, although closely related (like above, we can usually find different ways to produce the same image). 
And yes, the function $\rho$ is generally not a matrix, just something that spits out matrices.
