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).
**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 \(\mathbb{R}^3\).