Ideas of limit, approximation, and density can be used to prove some results about linear transforms on a real or complex vector space. For example, we can prove that any linear transform satisfies its own characteristics equation by observing that this is ture for diagonalisable matrices, and then argue by density of diagonalisable matrices and continuity of determinant.
Another example is the fact that two matrices $AB$ and $BA$ have the same characteristic polynomial. Again, assume at first that $A$ is invertible and use the approximation $A + \epsilon I$ when this is not the case.
However, it turns out that, many of such results, including the ones above, can be proven in a purely algebraic way, without reference to limit and continuity, and to be true for different other base fields without topology. (And, perhaps, giving better insight sometimes.)
So, my question is, are there any results in linear algebra that appear purely algebraic but must require some topology and limit to prove or very difficult to prove without topology?