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.

• 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,

Properties of Rank

• 0 ≤ rank(A) ≤ min⁡(m,n), A ∈ R_m×n
• If rank(A) = min⁡(m,n), the matrix is said to have full rank.
• For any matrix A, row rank = column rank = rank(A)
• A square matrix A is invertible   ⟺  rank(A) = n, the rank of an invertible matrix is equal to the order of the matrix.
• rank(AB) ≤ min⁡( rank(A), rank(B)).
• rank(A) = rank(A^⊤).
• rank(A+B) ≤ rank(A) + rank(B).
• rank(A^⊤A) = rank(A)
• Row operations do not change rank.

Properties of Nullity

• 0 ≤ nullity(A) ≤ n
• If rank(A)=n, then nullity(A)=0
• A square matrix A is invertible   ⟺  nullity(A)=0
• If nullity > 0, then Ax=0 has infinitely many nontrivial solutions.
• If nullity = 0, the only solution is the trivial one (x=0).

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

Applications 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.

Examples on Rank and Nullity

Some examples on rank and nullity are,

Example 1: Given Matrix

B = \begin{pmatrix} 1 & 1 & 0 & -2\\2 & 0 & 2 & 2 \\4 & 1 & 3 & 1 \\ \end{pmatrix}

Find the rank and nullity of B.

Solution:

B = \begin{pmatrix} 1 & 1 & 0 & -2\\2 & 0 & 2 & 2 \\4 & 1 & 3 & 1 \\ \end{pmatrix}

Using Row Transformation in matrix B,

R₂ → R₃ - 2R₂

B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\4 & 1 & 3 & 1 \\ \end{pmatrix}

Now, R₃ → R₃ - 4R₁

B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\0 & -3 & 3 & 9 \\ \end{pmatrix}

Now, R₃ → 3R₂ + R₃

B = \begin{pmatrix} 1 & 1 & 0 & -2\\0 & 1 & -1 & -3 \\0 & 0 & 0 & 0 \\ \end{pmatrix}

∴ 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

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

Find the rank of matrix A.

Solution:

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

Using Row Transformation in matrix A,

R₃ → R₃ + R₁

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

Now, R2 → R2 - 3R1

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

Now, R₃ → R₃ + R₂

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

r (A) = 2

∴ Rank of matrix A is 2.

Example 3: Given Matrix

D = \begin{pmatrix} 1 & 3\\0 & -2 \\5 & -1 \\-2 & 3 \\ \end{pmatrix}

Find the nullity of matrix D.

Solution:

D = \begin{pmatrix} 1 & 3\\0 & -2 \\5 & -1 \\-2 & 3 \\ \end{pmatrix}

Using Row Transformation in matrix D,

R₃ → R₃ - 5R₁

D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & -16 \\-2 & 3 \\ \end{pmatrix}

Now, R₄ → 2R₁ + R₄

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

Now, R₃ → -8R₂ + R₃

D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & 0 \\0 & 9 \\ \end{pmatrix}

Now, R₄ → 9R₂ + 2R₄

D = \begin{pmatrix} 1 & 3\\0 & -2 \\0 & 0 \\0 & 0 \\ \end{pmatrix}

Now, R₂ → -1/2 R₂

D = \begin{pmatrix} 1 & 3\\0 & 1 \\0 & 0 \\0 & 0 \\ \end{pmatrix}

r (D) = 2

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

∴ Nullity of matrix D is 0.

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