Eigenvectors
Eigenvectors are Vectors which remain in the same span when a linear transformation $A$ is applied to them. However, they can be scaled up or down. An eigenvector is represented algebraically as:
$$ Ax = \lambda I x $$
where $I$ is the identity matrix. This can also be rewritten as:
$$ (A - \lambda I)x = 0 $$
This linear system can have $1$ or $\infty$ solutions, since the $0$ vector is always a solution to the system(because of the 0 property of linear transformations).
The $\lambda$ here is called the eigenvalue of the eigenvector $x$.