Dimension

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  • 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.


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