Section 4.4 Rank of a Matrix
Deﬁnition: [latex]A[/latex] is a [latex]m × n[/latex] matrix. The column space, Col[latex]A[/latex], of [latex]A[/latex] is subspace spanned by columns of [latex]A[/latex]. The row space, Row[latex]A[/latex], of [latex]A[/latex] is the subspace of [latex]\mathbb{R}^{n}[/latex]? spanned by rows of [latex]A[/latex].
Fact: If [latex]A[/latex] is a reduced-echelon matrix, then the nonzero rows of [latex]A[/latex] form a basis of Row[latex]A[/latex] . The pivot columns of [latex]A[/latex] form a basis of Col[latex]A[/latex].
Definition: The rank of a matrix [latex]A[/latex], denoted by rank [latex]A[/latex], is the dimension of the column space of [latex]A[/latex].
Fact:
1. dim(Col [latex]A[/latex]) = dim(Row [latex]A[/latex]) = rank [latex]A[/latex].
2. rank [latex]A[/latex] = rank [latex]A^{T}[/latex].
The Rank Theorem: If a matrix [latex]A[/latex] has [latex]n[/latex] columns, then rank [latex]A[/latex] + dim Nul [latex]A[/latex] = [latex]n[/latex].
Example 1: If the subspace of all solutions of [latex]A\overrightarrow{x}=0[/latex]
has a basis consisting of three vectors and if [latex]A[/latex] is a [latex]5 \times 8[/latex] matrix, what is the rank of [latex]A[/latex]?
Exercise 1: What is the rank of a [latex]4 \times 7[/latex] matrix whose null space is four-dimensional?
Example 2: Suppose a [latex]4 \times 6[/latex] matrix [latex]A[/latex] has 4 pivot columns.
Is Col [latex]A=\mathbb{R}^{4}?[/latex]?
Is Nul [latex]A=\mathbb{R}^{2}[/latex]?
Explain your answer.
Exercise 2: Suppose a [latex]4 \times 7[/latex] matrix [latex]A[/latex] has 3 pivot columns.
Is Col [latex]A=\mathbb{R}^{3}[/latex]?
What is the dimension of Nul [latex]A[/latex]?
Explain your answer.
The Invertible Matrix Theorem:
Let [latex]A[/latex] be an [latex]n\times n[/latex] matrix.
Then the following statements are each equivalent to the statement that [latex]A[/latex] is an
invertible matrix:
(a) The columns of [latex]A[/latex] form a basis of [latex]\mathbb{R}^{n}[/latex].
(b) Col [latex]A=\mathbb{R}^{n}[/latex].
(c) dimCol [latex]A[/latex]=[latex]n[/latex].
(d) rank [latex]A[/latex]=[latex]n[/latex].
(e) Nul [latex]A\{0\}[/latex].
(f) dimNul [latex]A[/latex]=0.
Proof:
Theorem: The following are equivalent for an m x n matrix [latex]A[/latex]:
1. rank [latex]A[/latex]=n.
2. The rows of [latex]A[/latex] span [latex]\mathbb{R}^{n}[/latex].
3. The columns of [latex]A[/latex] are linearly independent in [latex]\mathbb{R}^{m}[/latex].
4. The n x n matrix [latex]A^{T}A[/latex] is invertible.
5. [latex]CA=I_{n}[/latex] for some n x m matrix [latex]C[/latex].
6. If [latex]A\overrightarrow{x}=0[/latex],[latex]\overrightarrow{x}[/latex] in [latex]\mathbb{R}^{n}[/latex], then [latex]\overrightarrow{x}=0[/latex].
Theorem: The following are equivalent for an [latex]m\times n[/latex] matrix [latex]A[/latex]:
1. rank [latex]A=m[/latex].
2. The columns of [latex]A[/latex] span [latex]\mathbb{R}^{m}[/latex].
3. The rows of [latex]A[/latex] are linearly independent in [latex]\mathbb{R}^{n}[/latex].
4. The [latex]m\times m[/latex] matrix [latex]AA^{T}[/latex] is invertible.
5. [latex]AC=I_{m}[/latex] for some [latex]n\times m[/latex] matrix [latex]C[/latex].
6. [latex]A\overrightarrow{x}=\overrightarrow{b}[/latex] is consistent for every [latex]\overrightarrow{b}[/latex] in [latex]\mathbb{R}^{m}[/latex].
Example 3: If [latex]A[/latex] is an [latex]m\times n[/latex] matrix and rank [latex]A=m[/latex],
show that [latex]m\leq n[/latex].
Exercise 3: If [latex]A[/latex] is an [latex]m\times n[/latex] matrix and columns of [latex]A[/latex] are linearly independent, show Nul [latex]A=\{\overrightarrow{0}\}[/latex].
GroupWork Example 1: True or False. Justify each answer:
a. Each line in [latex]\mathbb{R}^{n}[/latex] is a one-dimensional subspace of [latex]\mathbb{R}^{n}[/latex].
b. The dimension of Col [latex]A[/latex] is the number of pivot columns of [latex]A[/latex].
c. The dimensions of Col [latex]A[/latex] and Nul [latex]A[/latex] add up to the number of columns of [latex]A[/latex].
d. If a set of [latex]p[/latex] vectors spans a [latex]p[/latex]-dimensional subspace [latex]H[/latex] of [latex]\mathbb{R}^{n}[/latex], then these vectors form a basis for [latex]H[/latex].
e. The columns of an invertible [latex]n\times n[/latex] matrix form a basis for [latex]\mathbb{R}^{n}[/latex].
f. The dimension of Nul [latex]A[/latex] is the number of variables in the equation [latex]A\overrightarrow{x}=0[/latex]. The dimension of the column space of [latex]A[/latex] is rank [latex]A[/latex].
h. If [latex]H[/latex] is a [latex]p[/latex]-dimensional subspace of [latex]\mathbb{R}^{n}[/latex], then a linearly independent set of [latex]p[/latex] vectors in [latex]H[/latex] is a basis for [latex]H[/latex].
GroupWork 2: Suppose [latex]F[/latex] is a [latex]5\times5[/latex] matrix whose column space is not equal to [latex]\mathbb{R}^{5}[/latex].
What can you say about Nul [latex]F[/latex]?
GroupWork 3: Construct a nonzero [latex]3\times4[/latex] matrix [latex]A[/latex] such that dimNul [latex]A=2[/latex] and dimCol [latex]A=2[/latex].
a. Can a [latex]3\times4[/latex] matrix have independent columns? Independent rows? Explain.
b. If [latex]A[/latex] is [latex]4\times3[/latex] and rank [latex]A=2[/latex], can [latex]A[/latex] 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 [latex]3\times6[/latex] matrix have dimension [latex]2[/latex]? Explain.
GroupWork 5: Let [latex]A[/latex] be an [latex]n\times p[/latex] matrix whose column space is [latex]p[/latex]-dimensional. Explain why the columns of [latex]A[/latex] must be linearly independent.
GroupWork 6: Construct a [latex]4\times3[/latex] matrix with rank [latex]1[/latex].