linear algebra - Matrix Columns VS Rows - Mathematics Stack Exchange

i confused on geometric meaning of columns , rows in matrix. understanding 3blue1brown's linear algebra series elements of matrix column coordinates tip of single basis vector lands on @ end of linear transformation, while elements of matrix row coefficients of equation (if matrix augmented) or expression (if matrix not augmented). however, studying matrix subspaces, , appears both columns , rows can treated vector tip coordinates solve different subspaces.

is there second vector geometric interpretation of matrix elements of row coordinates on tip of single basis vector lands (which presumably identical applying original geometric interpretation matrix's transpose)? if so, geometric or conceptual difference between matrix columns , rows, justifies differences in how operate on them, , how relate them linear systems of equations?

actually, 3blue1brown gives interpretation well, though doesn't go deep enough particular aspect. in chapter 7, discusses duality - how linear transformations 1d line correspond specific vectors in space, , how when expressed matrices, transformation matrix column vector flipped on over side - i.e., converted row. these "row vectors" "dual vectors" of normal column vectors. , can consider linear transformations in terms of row vectors in similar fashion how videos talk them in terms of column vectors.

a basis provides way identify each vector in space specific dual vector. (in videos makes seem there 1 natural make assignment - because has natural basis, $\hat i, \hat j, \hat k$ uses. , covering more outside scope of trying do.) when transpose matrix, making use of vector-dual vector identification change transformation act on dual vectors instead of original vectors.

so, point is, can consider either columns or rows vectors. in either case, matrix converts vectors of same form other vectors of same form. when acting on columns, matrix multiplies in on left of column, while when acting on rows, multiplies in on right of row: $$\begin{bmatrix}a & b\\c & d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}w\\z\end{bmatrix}$$ $$\begin{bmatrix}p&q\end{bmatrix}\begin{bmatrix}a & b\\c & d\end{bmatrix} = \begin{bmatrix}r&s\end{bmatrix}$$