Rank and Nullity

Rank and Nullity are essential concepts in linear algebra, particularly in the context of matrices and linear transformations. They help describe the number of linearly independent vectors and the dimension of the kernel of a linear mapping.

In this article, we will learn what Rank and Nullity, the Rank-Nullity Theorem, and their applications, advantages, and limitations.

What is Rank and Nullity?

Rank of a Matrix is defined as the number of linearly independent row or column vectors of a matrix. It represents the number of non-zero eigenvalues of the matrix. The rank of a matrix is denoted as ρ(A).

Nullity of a Matrix is the dimension of its kernel, which is the number of independent solutions of the equation Ax = 0. It represents the number of zero eigenvalues of the matrix. The nullity of a matrix is denoted as N(A). For any matrix A of order 6×6 its rank and nullity are given below,

Nullspace

Nullspace of any matrix is defined as the solution associated with the system of homogenous equation AX = O where A is any real matrix of order, m × n.

Nullspace of A = { x ∈ Rⁿ | Ax = O}. Then the nullity of A is the dimension of the Nullspace of A.

Calculating Rank and Nullity

The rank and nullity of a matrix can be calculated using the following steps:

Row Reduction: Reduce the matrix to its row-reduced echelon form (RREF) using elementary row operations.

Counting Linearly Independent Vectors: Rank of a matrix is the number of linearly independent row or column vectors in the RREF.

Calculating Nullity: Nullity of a matrix is calculated by subtracting its rank from the total number of columns in the matrix.

Rank-Nullity Theorem

Rank-Nullity Theorem is a theorem in linear algebra that states that for a matrix M with x rows and y columns over a field, the rank of M and the nullity (the dimension of the kernel) of M sum to y.

For a matrix A of order n × n:

Rank of A + Nullity of A = Number of Columns in A = n

This can be generalized further to linear maps: if T: V → W is a linear map, then the dimension of the image of T plus the dimension of the kernel of T is equal to the dimension of V.

The theorem is useful in calculating either one by calculating the other instead, which is useful as it is often much easier to find the rank than the nullity (or vice versa).

There are several proofs of the rank-nullity theorem available; here is one such proof.

Rank-Nullity Theorem Proof

Statement: Let U and V be vector spaces over the field F and let T be a Linear Transformation (L.T.) from U into V. Suppose that U is finite-dimensional. Then, rank (T) + nullity (T) = dim U.

Proof:

Let N be a null space of T, then N is a subspace of U. Since U is finite-dimensional. Therefore, N is finite-dimensional.

Let dim N = nullity(T) = K and let {α₁, α₂,….., α_k} be a basis for N

∵ {α₁, α₂,….., α_k} is a linearly independent subset of U

∴ We can extend it to form a basis of U

Let dim U = n and let {α₁, α₂,….., α_k, α_k+1, α_k+2,……, α_n} be a basis of U

Vectors T(α₁),…., T(α_k), T(α_k+1), T(α_k+2),……., T(α_n) are in the range of T

We shall show that {T(α_k+1), T(α_k+2),…., T(α_n)} is a basis for the range of T

(I) First, we shall prove that the vectors T(α_k+1), T(α_k+2),….., T(α_n) span the range of T

Let β ∈ range of T, then ∃ α ∈ U such that T(α) = β

Now α ∈ U ⇒ ∃ a₁, a₂,……, a_n ∈ F such that

α = a₁α₁ + a₂α₂+……+ a_nα_n

⇒ T(α) = T(a₁α₁+a₂α₂+……+a_nα_n)

⇒ T(α) = T(a₁α₁+a₂α₂+……+a_kα_k+a_k+1α_k+1+……+a_nα_n)

⇒ β = a₁ T(α₁)+……+a_k T(α_k)+ a_k+1 T(α_k+1)+……+ a_n T(α_n)

⇒ β = a_k+1 T(α_k+1) + a_k+2 T(α_k+2) +…..+ a_n T(α_n)

[∵ α₁, α₂,….., α_k ∈ N ⇒ T(α₁) = 0,…., T(α_k) = 0]

∴ the vectors T(α_k+1), T(α_k+2),……, T(α_n) span the range of T.

(II) Now we shall show that the vectors T(α_k+1), T(α_k+2),……, T(α_n) are L.I.

Let c_k+1, c_k+2,…, c_n ∈ F such that

c_k+1T(α_k+1) + c_k+2T(α_k+2) +…..+ c_nT(α_n) = 0

⇒ T(c_k+1α_k+1 + c_k+2α_k+2 +……+ c_nα_n) = 0

⇒ c_k+1α_k+1, c_k+2α_k+2,….., c_nα_n ∈ null space of T, i.e., N

⇒ c_k+1α_k+1 + c_k+2α_k+2 +…….+ c_nα_n = b₁α₁ + b₂α₂ +…+ b_kα_k

for some b₁,b₂,….., b_k ∈ F.

[∵ Each vector in N can be expressed as a linear combination of the vectors α₁,……, α_n forming a basis of N]

⇒ b₁α₁ + b₂α₂ +…..+ b_kα_k – c_k+1α_k+1 – c_k+2α_k+2 -…….- c_nα_n = 0

⇒ b₁ = b₂ = …… = b_k = c_k+1 = c_k+2 = ……. = c_n = 0

[∵ α₁, α₂,…., α_k, α_k+1,……, α_n are linearly independent being basis of U]

⇒ Vectors T(α_k+1), T(α_k+2),….., T(α_n) are linearly independent.

∴ vectors T(α_k+1), T(α_k+2), …., T(α_n) form a basis of the range of a T.

∴ Rank T = Dim range of T = n – k

Hence proved

Advantages of Rank and Nullity

Aadvantages of understanding the rank and nullity of a matrix in linear algebra include:

• Simplifying Representation of Mathematical Problems: Rank and nullity provide a concise way to describe the structure of a matrix, making it easier to understand and solve problems involving linear algebra.

• Efficient Problem-Solving: By knowing the rank and nullity of a matrix, you can quickly identify the dimensions of the image and kernel of a linear transformation, which can help find the solution to a system of linear equations or determine the rank and nullity of a matrix.

• Providing Insights into Matrix Properties: The rank and nullity of a matrix can reveal information about the matrix, such as whether it is invertible or not, and can help in understanding the eigenvalues of the matrix.

• Rank-Nullity Theorem: This fundamental result in linear algebra states that the sum of the rank and nullity of a matrix equals the number of columns of the matrix. Understanding this theorem is crucial for solving a wide range of problems in linear algebra and forms the basis for various theoretical and practical applications.

Application of Rank and Nullity

The rank and nullity of a matrix have various applications in linear algebra, including:

• Solving Systems of Linear Equations: The rank and nullity of a matrix are used to determine the dimension of the kernel of a linear transformation, which in turn helps in solving systems of linear equations.

• Determining Dimension of Image and Kernel of a Linear Transformation: These concepts are essential for finding the dimensions of the image and kernel of a linear transformation, which is crucial in understanding the properties of the transformation.

• Matrix Theory: The rank and nullity of a matrix are fundamental in matrix theory, providing insights into the properties of the matrix, such as invertibility and eigenvalues.

Limitations of Rank and Nullity

While the rank and nullity of a matrix have numerous applications in linear algebra, there are some limitations to these concepts:

• Applicability: The rank and nullity of a matrix are only applicable to linear transformations and matrices. They may not be directly applied to non-linear transformations or matrices.

• Dimensionality: The rank and nullity of a matrix are defined for matrices with finite dimensions. They may not be directly applied to matrices with infinite dimensions or an infinite number of columns.

• Independence: The rank of a matrix is the number of linearly independent rows or columns. However, this concept may not be applicable when the matrix has dependent rows or columns, as the rank may not accurately represent the number of independent vectors.

• Invertibility: The rank of an invertible matrix is equal to the order of the matrix, and its nullity is equal to zero. This means that the rank and nullity of a matrix may not provide information about the invertibility of a matrix.

Conclusion of Rank and Nullity

These concepts are crucial for solving systems of linear equations, determining the dimensions of the image and kernel of a linear transformation, and simplifying the representation of mathematical problems.

Overall, the rank and nullity of a matrix are powerful tools that play a central role in various areas of linear algebra, providing valuable insights and enabling efficient problem-solving.

Related Artilces:
Matrices	Types of Matrices
Transpose of a Matrix	Inverse of Matrix

Examples on Rank and Nullity

Some examples on rank and nullity are,

Example 1: Given Matrix

[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\2 & 0 & 2 & 2 \\4 & 1 & 3 & 1 \\ \end{pmatrix}[/Tex]

Find the rank and nullity of B.

Solution:

[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\2 & 0 & 2 & 2 \\4 & 1 & 3 & 1 \\ \end{pmatrix}[/Tex]

Using Row Transformation in matrix B,

R₂ → R₃ – 2R₂

[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\4 & 1 & 3 & 1 \\ \end{pmatrix}[/Tex]

Now, R₃ → R₃ – 4R₁

[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\0 & -3 & 3 & 9 \\ \end{pmatrix}[/Tex]

Now, R₃ → 3R₂ + R₃

[Tex]B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\0 & 0 & 0 & 0 \\ \end{pmatrix}[/Tex]

∴ r (B) = 2.

n (B) = n (columns) – r (B) = 4 – 2 = 2.

∴ Rank of matrix B is 2 and the nullity of matrix B is 2.

Example 2: Given Matrix

[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\3 & 1 & 1 & 0 \\-1 & -5 & -1 & 8 \\ \end{pmatrix}[/Tex]

Find the rank of matrix A.

Solution:

[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\3 & 1 & 1 & 0 \\-1 & -5 & -1 & 8 \\ \end{pmatrix}[/Tex]

Using Row Transformation in matrix A,

R₃ → R₃ + R₁

[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\3 & 1 & 1 & 0 \\0 & -7 & -1 & 12 \\ \end{pmatrix}[/Tex]

Now, R2 → R2 – 3R1

[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\0 & 7 & 1 & -12 \\0 & -7 & -1 & 12 \\ \end{pmatrix}[/Tex]

Now, R₃ → R₃ + R₂

[Tex]A = \begin{pmatrix} 1 & -2 & 0 & 4\\0 & 7 & 1 & -12 \\0 & 0 & 0 & 0 \\ \end{pmatrix}[/Tex]

r (A) = 2

∴ Rank of matrix A is 2.

Example 3: Given Matrix

[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\5 & -1 \\-2 & 3 \\ \end{pmatrix}[/Tex]

Find the nullity of matrix D.

Solution:

[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\5 & -1 \\-2 & 3 \\ \end{pmatrix}[/Tex]

Using Row Transformation in matrix D,

R₃ → R₃ – 5R₁

[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & -16 \\-2 & 3 \\ \end{pmatrix}[/Tex]

Now, R₄ → 2R₁ + R₄

[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & -16 \\0 & 9 \\ \end{pmatrix}[/Tex]

Now, R₃ → -8R₂ + R₃

[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & 0 \\0 & 9 \\ \end{pmatrix}[/Tex]

Now, R₄ → 9R₂ + 2R₄

[Tex]D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & 0 \\0 & 0 \\ \end{pmatrix}[/Tex]

Now, R₂ → -1/2 R₂

[Tex]D = \begin{pmatrix} 1 & 3\\0 & 1 \\0 & 0 \\0 & 0 \\ \end{pmatrix}[/Tex]

r (D) = 2

n (D) = n (columns) – r (D) = 2 – 2 = 0.

∴ Nullity of matrix D is 0.

Frequently Asked Questions on Rank and Nullity

What is the Rank of a Matrix?

Rank of the matrix is defined as the number of linearly independent row or column vectors of a matrix.

What is the Nullity of a Matrix?

Nullity of the Matrix is defined as the dimension of the nullspace or kernel of the given matrix.

What is the Rank and Nullity Theorem?

Rank and Nullity Theorem for Matrices says that for any matrix A of order m by n,

rank(A) + nullity(A) = n = number of columns in A

How is Rank of a Matrix Determined?

Rank of a matrix is determined by counting the number of linearly independent rows or columns after applying row transformations and forming an upper triangular matrix.

What is Significance of Nullity of a Matrix in Linear Algebra?

Nullity of a matrix represents the dimension of its kernel, providing information about the number of independent solutions to the homogeneous equation Ax=0.

How are Rank and Nullity related in Context of Rank-Nullity theorem?

Rank-Nullity theorem states that the sum of the rank and nullity of a matrix is equal to the number of columns of the matrix.

What are Advantages of Understanding Rank and Nullity of a Matrix?

Understanding rank and nullity simplifies mathematical problem representation, enables efficient problem-solving, and provides insights into matrix properties.

What is the Nullity of an Invertible Matrix?

Nullity of an Invertible Matrix is Zero.