Two \((n,k)\) linear codes are said to be isomorphic if there is a permutation of the coordinates that maps one code onto the other. The intersection of two isomorphic \((n,k)\) codes is an \((n,k')\) code, where clearly \(\max\{0,2k-n\}\leq k' \leq k\). In this paper, the problem of finding all attainable values of \(k'\), called intersection numbers, is considered. This problem was already solved for binary Hamming codes by \textit{T. Etzion} and \textit{A. Vardy} [SIAM J. Discrete Math. 11, 205-223 (1998; Zbl 0908.94035)]. It is shown how attainable intersection numbers can be deduced from the structure of the generator matrix of a code. This result is used to solve the intersection problem for cyclic codes and, with some exceptions, for extended cyclic codes and MDS codes.