I'm working on restoring my linear algebra knowledge from a few years ago by walking through a few simple examples to rebuild my intuitions, and I stumbled on an interesting problem.
I was trying to find the eigenvectors and eigenvalues of $\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix}$ and it seems that the constraints defining the eigenvectors allow for any vector in the span composed of two lines:
$$\vec{v} = \begin{bmatrix} n \\ 2n \end{bmatrix} \forall \ n$$ $$\vec{v} = \begin{bmatrix} n \\ -n \end{bmatrix} \forall \ n$$
But this presents an interesting question: For all n in what number space?
When tasked with finding the eigenvectors of a matrix, would there be some implicit assumption we're talking about the reals ($\mathbb{R}$), indicating an eigenvector along a line in 2-dimensional space, or could we be talking about the complex number space ($\mathbb{C}$), indicating a plane in 3-dimensional space?
Since this is Linear Algebra, the eigenspaces must be vector spaces. So, $\mathbb N$ cannot possibly be an option here.
Is it $\mathbb R$ or is it $\mathbb C$? It could also be $\mathbb Q$. And there are other possibilities. It all depends upon how the problem is stated and also upon its context.