Null Space Calculator
Null Space Analysis
Dimension of Null Space:
Basis Vectors:
What is a Null Space?
The null space (or kernel) of a matrix A is the set of all vectors x that satisfy the equation Ax = 0. It represents all solutions to the homogeneous system of linear equations.
Understanding Null Space
Key Concepts
- Basis vectors form a spanning set
- Dimension indicates number of free variables
- Related to matrix rank and linear independence
- Important in linear transformations
Applications
- Linear Systems of Equations
- Linear Transformations
- Vector Space Analysis
- Engineering Systems
- Computer Graphics
How to Use the Calculator
- Enter matrix dimensions (rows and columns)
- Input matrix elements
- Click "Find Null Space" to calculate
- Review basis vectors and dimension
Frequently Asked Questions
What does null space dimension mean?
The dimension of the null space equals the number of linearly independent solutions to the homogeneous system. It's also equal to the number of free variables in the system.
How is null space calculated?
The null space is found by reducing the matrix to row echelon form, identifying free variables, and expressing the solution in terms of these variables.
Why is null space important?
Null space helps understand the structure of linear transformations, solve systems of equations, and analyze linear independence of vectors.