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 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
Contact Us