The Rank of a Matrix:
Rank of a Matrix:
Recall:
Let
[pic].
The i’th row of A is
[pic],
and the j’th column of A is
[pic]
Definition of row space and column space:
[pic],
which is a vector space under standard matrix addition and scalar multiplication, is referred to as the row space. Similarly,
[pic],
which is also a vector space under standard matrix addition and scalar multiplication, is referred to as the column space.
Definition of row equivalence:
A matrix B is row equivalent to a matrix A if B result from A via elementary row operations.
Example:
Let
[pic]
Since
[pic],
[pic],
[pic],
[pic] are all row equivalent to [pic].
Important Result:
If A and B are two [pic] row equivalent matrices, then the row spaces of A and B are equal.
How to find the bases of the row and column spaces:
Suppose A is a [pic] matrix. Then, the bases of the row and column spaces can be found via the following steps.
Step 1:
Transform the matrix A to the matrix in reduced row echelon form.
Step 2:
• The nonzero rows of the matrix in reduced row echelon form form a basis of the row space of A.
• The columns corresponding to the ones containing the leading 1’s form a basis. For example, if [pic] and the reduced row echelon matrix is
[pic],
then the 1’st, the 3’rd, and the 4’th columns contain a leading 1 and
thus [pic] form a basis of the column
space of A.
Note:
To find the basis of the column space is to find to basis for the vector space [pic]. Two methods introduced in the previous section can also be used. The method used in this section is equivalent to the second method in the previous section.
Example:
Let
[pic].
Find the bases of the row and column spaces of A.
[solution:]
Step 1:
Transform the matrix A to the matrix in reduced row echelon form,
[pic]
Step 2:
• The basis for the row space is
[pic]
• The columns corresponding to the ones containing the leading 1’s are the 1’st, the 2’nd, and the 4’th columns. Thus,
[pic]
form a basis of the column space.
Definition of row rank and column rank:
The dimension of the row space of A is called the row rank of A and the dimension of the column space of A is called the column rank of A.
Example (continue):
Since the basis of the row space of A is
[pic],
the dimension of the row space is 3 and the row rank of A is 3. Similarly,
[pic]
is the basis of the column space of A. Thus, the dimension of the column space is 3 and the column rank of A is 3.
Important Result:
The row rank and column rank of the [pic] matrix A are equal.
Definition of the rank of a matrix:
Since the row rank and the column rank of a [pic]matrix A are equal, we only refer to the rank of A and write [pic].
Important Result:
If A is a [pic] matrix , then
[pic]
Example:
[pic] and [pic].
Since
[pic]
is a basis of column space and thus [pic]. The solutions of [pic] are
[pic].
Thus, the solution space (the null space) is
[pic].
Then, [pic] and [pic] are the basis of the null space. and [pic].
Therefore,
[pic].
Important Result:
Let A be an [pic] matrix.
• A is nonsingular if and only if [pic].
• [pic]
• [pic]
Important Result:
Let A be an [pic] matrix. Then,
[pic] has a solution [pic]
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