Two problems that I have myself come across are:
- The invertibility of a matrix can change. For example,the integer matrix $$\left[\begin{matrix}4 & 1\\2 & 2\end{matrix}\right]$$ is invertible over the real field because its determinant is nonzero (it is six). Considering the same matrix over the field $\mathbb{F}_3$ (with characteristic three), the determinant is zero and the same matrix is now singular. In this case the matrix should be written as $$\left[\begin{matrix}1 & 1\\2 & 2\end{matrix}\right]$$ and it is easy to see that one row is the multiple of the other.
- This one is the shocking one that you can have a nonzero vector with norm zero. For example, consider $$x=[1,1,1]$$ which is obviously a nonzero vector over the field $\mathbb{F}_3$ but using the usual inner-product and norm definitions $$||x||^2=|<x,x>|^2=0.$$ So this nonzero vector is orthogonal to itself. Over the real and the complex field, only the zero vector is orthogonal to itself. Now just see how many of the standard linear algebra algorithms fail because of this fact. For one, QR decomposition won't work because Gram-Scmidt won't work here. Something as simple as normalizing a vector would fail.
