Some false beliefs in linear algebra:
If two operators or matrices A, B commute, then they are simultaneously diagonalisable. (Of course, this overlooks the obvious necessary condition that each of A, B must first be individually diagonalisable. Part of the problem is that this is not an issue in the Hermitian case, which is usually the case one is most frequently exposed to.)
The operator norm of a matrix is the same as the magnitude of the most extreme eigenvalue. (Again, true in the Hermitian or normal case, but in the general case one has to deal instead with the spectral radius or singular values.)
The singular values of a matrix are the absolute values of the eigenvalues of the matrix. (Closely related to the previous false belief.)
If a matrix has distinct eigenvalues, then one can find an orthonormal eigenbasis. (The normality is only possible when the matrix is, well, normal.)
A matrix is diagonalisable if and only if it has distinct eigenvalues. (Only the "if" part is true. The identity matrix and zero matrix are blatant counterexamples, but I have seen this false belief persist remarkably well nonetheless.)
If L: X -> Y is a continuous linear transformation that is surjective (i.e. Lu=f is always solvable for any data f in Y), and X and Y are Banach spaces then it has a continuous linear inverse. (This is subtle. Zorn's lemma gives a linear inverse; the open mapping theorem (morally) gives a continuous inverse. But getting an inverse that is simultaneously continuous and linear is not always possible!)