Rank

1
Rank
2
Rank
3
Rank
4
Rank

• Equivalent descriptions of rank
• The rank of an m x n matrix A may be described in
  several ways
• The dimension of the column space of A
• The number of pivotal positions in A
• The maximum number of linearly independent
  columns in A
• The dimension of the row space of A
• The maximum number of linearly independent rows
  in A
• The number of nonzero rows in an echelon form of
  A
• The maximum nbr of columns in an invertible
  submatrix of A

5
Rank
Effects of Elementary Row Operations -Row
operations do not affect the linear dependence
relations among the columns. -row operations
usually change the column space -row operations
never change the row space. Row operations never
change the null space.
6
Rank
m x n Matrix A
7
Rank
n x n Matrix A
8
Rank
n x p Matrix A
9
Rank
10
Rank
11
Rank
12
Rank