Matrix Input
Results
Reduced Row Echelon Form (RREF)
Rank & Nullity
Kernel Basis Vectors
How to Use the Kernel Calculator
Unlock the solution to Ax = 0 in three simple steps. Our intuitive tool makes finding the null space of any matrix straightforward and insightful.
Input Matrix
Use the interactive grid or paste spreadsheet data. The calculator handles real, fractional, and complex numbers.
Calculate
Click "Find Kernel" to initiate Gaussian elimination and automatically solve for the basis of the null space.
Analyze Results
Review the RREF matrix, rank, nullity, and the resulting basis vectors, all clearly formatted for easy understanding.
The Rank-Nullity Theorem Explained
This fundamental theorem of linear algebra provides a beautiful and powerful connection between the size of a matrix and its core subspaces. Our tool visualizes this theorem for every calculation you make.
What is Rank?
The rank of a matrix A is the dimension of its column space (or row space). It represents the number of linearly independent columns, or "pivot columns," and indicates the dimension of the image of the linear transformation.
What is Nullity?
The nullity of a matrix A is the dimension of its kernel (null space). It represents the number of "free variables" in the system Ax = 0 and tells you how many dimensions are "collapsed" to zero by the transformation.
Core Features & Real-World Applications
This calculator is more than just a solver; it's a platform for exploring the concepts that power modern science and technology.
Key Features
- Advanced RREF Engine: Employs Gaussian elimination to handle any matrix, including those with complex numbers.
- Complete Solution Output: Provides the RREF, rank, nullity, and basis vectors for a full picture.
- Versatile Data Handling: Supports real, fractional, and complex numbers (e.g.,
5-3i) in all calculations. - Intuitive Interface: Clean, responsive design with clear instructions and error handling.
Applications
- Computer Graphics: Determines lines or planes that are invisible from a certain perspective.
- Control Theory: Helps analyze the stability and controllability of linear systems.
- Error-Correcting Codes: The null space of a parity-check matrix defines the set of valid codewords.
- Chemistry: Used to balance complex chemical equations by solving stoichiometric systems.
Frequently Asked Questions
This is called a trivial kernel. It means the nullity of your matrix is 0. Geometrically, this is excellent news for many applications! It signifies that the linear transformation is one-to-one (injective), meaning no two distinct vectors are mapped to the same output vector.
No, the basis itself is not unique. A vector space can have infinitely many different bases. However, all bases for a given vector space (like the kernel) will have the same number of vectors. That number is the dimension of the space, which our calculator gives you as the "Nullity."
Kernel and Null Space refer to the exact same thing: the set of vectors x such that Ax = 0. The terms are used interchangeably. Nullity is the *dimension* of that space—in other words, the number of vectors in its basis.
