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.
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.
Table of Content
- What is Rank and Nullity?
- Calculating Rank and Nullity
- Rank-Nullity Theorem
- Rank-Nullity Theorem Proof
- Advantages of Rank and Nullity
- Application of Rank and Nullity
- Limitations of Rank and Nullity