Rank and Nullity

Various Subspaces

Row Space, Column Space, Null Space

Let be an matrix over . The row space of , denoted , is the subspace of spanned by the row vectors of . The column space of , denoted , is the subspace of spanned by the column vectors of . The null space of , denoted or , is the subspace of consisting of all solutions to the homogeneous equation .

We immediately have the following characterization of solutions to linear equations:

Proposition

The equation is consistent iff . If is a particular solution to , then the complete solution set is .

Rank

Theorem

The row space and column space of a matrix have the same dimension (as subspaces of ).

Rank & Nullity

The rank of a matrix is the dimension of or . The nullity of a matrix is the dimension of .

Theorem

For an matrix , the followings are equivalent:

1. is invertible;
2. has only the trivial solution;
3. ;
4. is a product of elementary matrices;
5. is consistent for every ;
6. has exactly one solution for every ;
7. ;
8. The column vectors of are linearly independent;
9. The row vectors of are linearly independent;
10. The column vectors of span ;
11. The row vectors of span ;
12. The column vectors of is a basis for ;
13. The row vectors of is a basis for ;
14. ;
15. .