Section 4.4 Rank of a Matrix

Definition:  A is a m × 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 Rn spanned by rows of A.

Fact: If A is a reduced-echelon matrix, then the nonzero rows of basis of RowA. The pivot columns of A are a basis of ColA.

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 x 8 matrix, what is the rank of $A$?

Exercise 1: What is the rank of a 4 x 7 matrix whose null space is two-dimensional?

Example 2: Suppose a 4 x 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 x 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

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$A=\{\overrightarrow{0}\}$.

Group Work 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 $H$.

Group Work 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 $F$ ?

Group Work 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

Group Work 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

Group Work 6: Construct a $4\times 3$ matrix with rank $1$