An elementary proof using matrix theory is given for the following criterion: if "F"/"K" and "L"/"K" are field extensions, with "F" and "L" both contained in a common extension field, then "F" and "L" are linearly disjoint over "K" if (and only if) some "K"-vector space basis of "F" is linearly independent over "L." The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra.