The row space of a matrix is a vector space

**Theorem:** The *row space* of a matrix \(A\) (the span of its row vectors) is a vector space --- specifically, a subspace of \(\mathbb{F}^n\) where \(n\) is the number of columns.

Notation: \(\text{row}(A)\) or \(\text{rowspace}(A)\).

Since the row space is defined as a span, it is automatically a subspace (spans are always subspaces).

**Key fact:** The dimension of the row space equals the *rank* of the matrix. The row rank equals the column rank.

**Example:** For \(A = \begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 6 \end{pmatrix}\), the row space is spanned by \((1,2,3)\) alone (since row 2 = 2 times row 1), so it is a 1-dimensional subspace of \(\mathbb{R}^3\).Copy