Sensitive question on the definition of eigenvalues and eigenvectors

Question

I'm asking a sensitive question.

Suppose, you are a student who is taking an exam on linear algebra. Suppose you're encountering a question asking you to compute the eigenvalues and eigenvectors for $A\in M_n(\Bbb R)$. Such problem is stated as below:

Q: Compute the eigenvalues and eigenvectors for $\begin{bmatrix}3&2\\2&0\end{bmatrix}$.

Now, after some computation, you gain the eigenvalues are $-1,~4$. Next you wish to get the eigenvectors.

For $\lambda=-1:$ we are going to solve the linear system $\begin{bmatrix}3+1&2\\2&0+1\end{bmatrix}$. Then $\begin{cases}x=-t\\y=2t\end{cases}$.

However, what should the range of $t$ be here? Of course by definition we can't take $t=0$. But what is the exact range we should write?

$\begin{cases}x=-t\\y=2t\end{cases}(t\in\Bbb R\setminus\{0\})$?

Or $\begin{cases}x=-t\\y=2t\end{cases}(t\in\Bbb C\setminus\{0\})$?

Or $\begin{cases}x=-t\\y=2t\end{cases}(t\in\Bbb Q(\sqrt{2})\setminus\{0\})$

Answer 1

The eigenvectors with eigenvalue $-1$ are the vectors of the type $(-t,2t)$ with $t\in\mathbb{R}\setminus\{0\}$. But the eigenspace which corresponds to the eigenvalue $-1$ is$$\left\{(-t,2t)\,\middle|\,t\in\mathbb R\right\}.$$Of course, in the general case we should replace $\mathbb R$ with the field $F$ that we're working with.

Answer 2

It depends on the field $F$ you're working over... You get all scalar multiples $t \begin{bmatrix}-1\\2 \end{bmatrix}$ for $t\in F\setminus \{0\}$...

Answer 3

You used the argument that $A∈M_n(\Bbb R)$, so you're talking about a linear mapping of the form $\Bbb R^n \to \Bbb R^n$. The eigenspace being a sub-space of the co-domain, you clearly need $t \in \Bbb R$. If we were dealing with a co-domain of vectors over $\Bbb C$, your argument would be valid. Keep in mind we're working with $M_n(\Bbb R)$ rather than $M_n(\Bbb C)$ or $M_n(\Bbb C, \Bbb R)$.