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: ( ext{row}(A)) or ( ext{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 = egin{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).