Contravariant and Covariant Objects in Matrix Notation

For many years when and since I was in college, I wondered whether it might be possible to consistently represent contravariant & covariant objects using vector & matrix notation. In particular, when I learned about the idea of covariant representations of [invariant] vectors being duals to contravariant representations of [invariant] vectors, meaning that if a contravariant representation of a [invariant] vector can be seen as a column vector, then a covariant representation of a [invariant] vector can be seen as a row vector, I wondered how it would be possible to represent the fully covariant metric tensor as a metric tensor if it multiplies a contravariant representation of a [invariant] vector (i.e. a column vector) to yield a covariant representation of a [invariant] vector (i.e. a row vector), especially as traditionally in linear algebra, a matrix acting on a column vector yields another column vector (while transposition, though linear in the sense of respecting addition and scalar multiplication, cannot be represented simply as the action of another matrix). At various points, I've wondered if this means that fully contravariant or fully covariant representations of multi-index tensors should be represented as columns of columns or rows of rows, and I've tried to play around with these ideas more. This post is not the first to explore such ideas even online, as I came across notes online by Viktor T. Toth [LINK], but this post is my attempt to flesh out these ideas further. Follow the jump to see more. Throughout this post, I will work with the notation of 2 spatial indices, in which the fully covariant representation of the metric tensor \( g_{ij} = \vec{e}_{i} \cdot \vec{e}_{j} \) might not be Euclidean, where indices will use English letters \( i, j, k, \ldots \in \{1, 2\} \), where superscripts do not imply exponents, and where multiple superscripts do not imply single numbers (for example, \( g_{12} \) is the fully covariant component of the metric tensor with first index 1 and second index 2, not the covariant component at index 12 of a single-index tensor (vector)); extensions to spacetime (where the convention is to use indices labeled by Greek letters) and in particular to 3 spatial + 1 temporal dimensions are trivial. Additionally, Einstein summation will be assumed, and all tensors (including vectors & scalars) are assumed to be real-valued. Finally, I will do my best to ensure that when indices are raised or lowered, the ordering of indices is clear (as examples, distinguishing \( T^{i}_{\, j} \) from \( T_{i}^{\, j} \) instead of ambiguously using \( T^{i}_{j} \) or \( T^{j}_{i} \)), but this will depend on the quality of LaTeX rendering in this post.