row and column spaces

Let \[F\] be a field. The column space of an \[m\times n\] matrix with components (or elements) from \[F\] is a linear subspace of the \[m\]-dimensional space \[F^{m}\] (e.g. if \[F\] is the field of real numbers, then \[F^{m}=\mathbb{R}^{m}\]). The dimension of the column space is known as the rank of the matrix and is at most the minimum between \[m\] and \[n\].

The definition for row space is pretty similar. Since every row is an element of the space \[F^{n}\], the row space is a linear subspace of \[F^{n}\]. The dimension of the row space is called the row rank (which is also equivalent to the column rank) of the matrix and is at most the minimum between \[m\] and \[n\].

Assume we have a matrix \[A=\begin{pmatrix} 1&4&7&19\\11&2&6&-2\\1&5&8&2 \end{pmatrix}\], to find it's column space, we chop it into four columns, \[c_{1}=\begin{pmatrix} 1\\11\\1 \end{pmatrix}\], \[c_{2}=\begin{pmatrix} 4\\2\\5 \end{pmatrix}\], \[c_{3}=\begin{pmatrix} 7\\6\\8 \end{pmatrix}\] and \[c_{4}=\begin{pmatrix} 19\\-2\\2 \end{pmatrix}\]. Then, the column space of \[A\] would be \[\text{span}(\left\{ c_{1},c_{2},c_{3},c_{4} \right\})\]. The row space of \[A\] can be demonstrated in a similar fashion.

In general, row spaces and column spaces can be very different. For non-square matrices, they will be in different dimension ambient spaces (although the dimension of the subspace spanned by each will be the same as mentioned above).