Row Space

The row space of an $m\times n$ matrix $\mathbf{A}$, denoted $\text{Row}(\mathbf{A})$, is the span of its $m$ rows, a subspace of ${\mathbb{R}}^n$; equivalently, $\text{Row}(\mathbf{A})=\text{Col}({\mathbf{A}}^{\top })$, the column space of the transpose. A fundamental (and non-obvious) theorem of linear algebra states that the dimension of the row space always equals the dimension of the column space, both equal to $\text{rank}(\mathbf{A})$, even though the row space lives in ${\mathbb{R}}^n$ and the column space lives in ${\mathbb{R}}^m$, generally different spaces when $\mathbf{A}$ is not square. This shared dimension is precisely why 'rank' can be discussed as a single number describing a matrix, without needing to separately track a distinct row rank and column rank.

• The row space and column space of a non-square matrix are generally distinct subspaces living in different-dimensional ambient spaces, ${\mathbb{R}}^n$ for the row space and ${\mathbb{R}}^m$ for the column space, yet the fact that their dimensions always coincide (both equal to rank) is one of the foundational, somewhat surprising results in elementary linear algebra.
• The row space of a matrix equals the column space of its transpose, $\text{Row}(\mathbf{A})=\text{Col}({\mathbf{A}}^{\top })$ -- this equivalence means every property or computational technique developed for column spaces can be applied to row spaces simply by transposing the matrix first.
• For a symmetric matrix, the row space and column space are literally the same subspace, since $\mathbf{A}={\mathbf{A}}^{\top }$ -- this is one more simplification, alongside real eigenvalues and orthogonal eigenvectors, that symmetric matrices such as covariance matrices enjoy compared to general matrices.