# Dimension

• Of a set: the dimension of the smallest affine space that contains the set. The dimension of an affine space is $LaTeX: k$ if the maximum number of affinely independent points in the set is $LaTeX: k+1$. Of particular interest is the column space of a matrix $LaTeX: A$,
$LaTeX: \mbox{col}(A) = \{x: x=Ay \mbox{ for some } y \in \mathbb{R}^n\}$.
The dimension of this is the rank of $LaTeX: A$. Another is the row space of $LaTeX: A$,
$LaTeX: \mbox{row}(A) = \{x: x=A^Tu \mbox{ for some } u \in \mathbb{R}^m\}$.
The dimension of this is also the rank of $LaTeX: A$.
• Of an expression: the units of measurement (e.g., tons of apples, meters of rope, hours of labor). In a relation, such as $LaTeX: y = ax + b$, the units of $LaTeX: y$, $LaTeX: ax$ and $LaTeX: b$ must all be the same. Dimensional analysis is concerned with determining such consistency and inferring whatever units are missing.