MATH125 Lecture Notes - Lecture 22: Row And Column Spaces

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Nikita

Document Summary

In practice most of subspaces occur as the following subspaces cor- responding to matrices. Let a be a matrix of size m n. the column subspace of a matrix a is the set of all linear combinations of the columns of a. From our examples we know that it is a vector subspace in rm and it is denoted by col a. It follows from de nition that if a = [a1 a2 an] where a1, . , an are columns of a then col a is the same as span {a1, . The vector b is in col a if and only if it can be written as a linear combination of columns of a, i. e. if and only if the equation. It follows that our equation has a solution (in fact it has in nitely many solutions because we have a free variable), hence b col a.