Say we have a morphism $f$ from some $\mathbb{k}$-algebra $A$ to $\mathbb{k}$, but we also know that $A$ is a vectorspace i.e. we can define a basis on $A$. Then clearly to look at the image of $A$ we can just look at what $f$ does to the basis of $A$. So say the basis elements of $A$ are written as $e_i$, so we look at $f(e_i) : = \lambda_i$, where each $\lambda_i \in \mathbb{k}$.
What can we say about the set of images of each basis element in $\mathbb{k}$, so that $f$ is an algebra morphism? Other than dimensions being equal, I can't think of any other specific properties.