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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