Matrix notation involves using a specific basis for the vector space, resulting in matrices represented as a two-dimensional array of coefficients. In contrast, abstract matrix notation does not rely on a chosen basis, allowing for a more generalized representation. For instance, in matrix notation, the operation of multiplying a vector by a matrix is expressed with indices, while in abstract notation, it is simplified to a more straightforward form. This distinction highlights the level of specificity versus generality in mathematical expressions. Understanding these differences is crucial for clarity in computational methods work.