Section 4.4 Rank of a Matrix

Definition:  $A$ is a $m\times n$ matrix. The column space, Col$A$, of $A$ is subspace spanned by columns of $A$. The row space, Row$A$, of $A$ is the subspace of ${\mathbb{R}}^n$? spanned by rows of $A$.

Fact: If $A$ is a reduced-echelon matrix, then the nonzero rows of $A$ form a basis of Row$A$ . The pivot columns of $A$ form a basis of Col$A$.

 

Definition: The rank of a matrix $A$, denoted by rank $A$, is the dimension of the column space of $A$.

Fact:

1. dim(Col $A$) = dim(Row $A$) = rank $A$.

2. rank $A$ = rank $A^T$.

 

The Rank Theorem: If a matrix $A$ has $n$ columns, then rank $A$ + dim Nul $A$ = $n$.

 

Example 1: If the subspace of all solutions of $A\overrightarrow{x}=0$
has a basis consisting of three vectors and if $A$ is a $5\times 8$ matrix, what is the rank of $A$?

 

Exercise 1: What is the rank of a $4\times 7$ matrix whose null space is four-dimensional?

 

Example 2: Suppose a $4\times 6$ matrix $A$ has 4 pivot columns.

Is Col $A={\mathbb{R}}^4?$?

Is Nul $A={\mathbb{R}}^2$?

Explain your answer.

 

 

Exercise 2: Suppose a $4\times 7$ matrix $A$ has 3 pivot columns.
Is Col $A={\mathbb{R}}^3$?

What is the dimension of Nul $A$?

Explain your answer.

 

The Invertible Matrix Theorem:
Let $A$ be an $n\times n$ matrix.
Then the following statements are each equivalent to the statement that $A$ is an
invertible matrix:

(a) The columns of $A$ form a basis of ${\mathbb{R}}^n$.

(b) Col $A={\mathbb{R}}^n$.

(c) dimCol $A$=$n$.

(d) rank $A$=$n$.

(e) Nul $A\{0\}$.

(f) dimNul $A$=0.

 

Proof:

 

Theorem: The following are equivalent for an m x n matrix $A$:

1. rank $A$=n.

2. The rows of $A$ span ${\mathbb{R}}^n$.

3. The columns of $A$ are linearly independent in ${\mathbb{R}}^m$.

4. The n x n matrix $A^{T}A$ is invertible.

5. $CA=I_n$ for some n x m matrix $C$.

6. If $A\overrightarrow{x}=0$,$\overrightarrow{x}$ in ${\mathbb{R}}^n$, then $\overrightarrow{x}=0$.

 

 

Theorem: The following are equivalent for an $m\times n$ matrix $A$:

1. rank $A=m$.

2. The columns of $A$ span ${\mathbb{R}}^m$.

3. The rows of $A$ are linearly independent in ${\mathbb{R}}^n$.

4. The $m\times m$ matrix $A{A}^T$ is invertible.

5. $AC=I_m$ for some $n\times m$ matrix $C$.

6. $A\overrightarrow{x}=\overrightarrow{b}$ is consistent for every $\overrightarrow{b}$ in ${\mathbb{R}}^m$.

 

 

Example 3: If $A$ is an $m\times n$ matrix and rank $A=m$,
show that $m\le n$.

 

 

Exercise 3: If $A$ is an $m\times n$ matrix and columns of $A$ are linearly independent, show Nul $A=\{\overrightarrow{0}\}$.

 

GroupWork Example 1: True or False. Justify each answer:

a. Each line in ${\mathbb{R}}^n$ is a one-dimensional subspace of ${\mathbb{R}}^n$.

b. The dimension of Col $A$ is the number of pivot columns of $A$.

c. The dimensions of Col $A$ and Nul $A$ add up to the number of columns of $A$.

d. If a set of $p$ vectors spans a $p$-dimensional subspace $H$ of ${\mathbb{R}}^n$, then these vectors form a basis for $H$.

e. The columns of an invertible $n\times n$ matrix form a basis for ${\mathbb{R}}^n$.

f. The dimension of Nul $A$ is the number of variables in the equation $A\overrightarrow{x}=0$. The dimension of the column space of $A$ is rank $A$.

h. If $H$ is a $p$-dimensional subspace of ${\mathbb{R}}^n$, then a linearly independent set of $p$ vectors in $H$ is a basis for $H$.

 

GroupWork 2: Suppose $F$ is a $5\times 5$ matrix whose column space is not equal to ${\mathbb{R}}^5$.
What can you say about Nul $F$?

 

GroupWork 3: Construct a nonzero $3\times 4$ matrix $A$ such that dimNul $A=2$ and dimCol $A=2$.

a. Can a $3\times 4$ matrix have independent columns? Independent rows? Explain.

b. If $A$ is $4\times 3$ and rank $A=2$, can $A$ have independent columns? Independent rows? Explain.

c. Can a non-square matrix have its rows independent and its columns independent? Explain.

d. Can the null space of a $3\times 6$ matrix have dimension $2$? Explain.

 

GroupWork 5: Let $A$ be an $n\times p$ matrix whose column space is $p$-dimensional. Explain why the columns of $A$ must be linearly independent.

 

GroupWork 6: Construct a $4\times 3$ matrix with rank $1$.