I am confused over the geometric meaning of columns and rows in a matrix. My understanding from 3Blue1Brown's Linear Algebra series was that the elements of a matrix column are the coordinates which the tip of a single basis vector lands on at the end of the linear transformation, while the elements of a matrix row are the coefficients of an equation (if the matrix is augmented) or expression (if the matrix is not augmented). However, I am now studying matrix subspaces, and it appears that both columns and rows can be treated as vector tip coordinates to solve for different subspaces.
Is there a second vector geometric interpretation of a matrix where the elements of a row are the coordinates on which the tip of a single basis vector lands (which would presumably be identical to applying the original geometric interpretation to the matrix's transpose)? If so, what is the geometric or conceptual difference between matrix columns and rows, which justifies the differences in how we operate on them, and how we relate them to linear systems of equations?