Null Space Calculator
Find the null space (kernel) of a matrix and calculate its nullity with step-by-step solutions
Matrix Input
Enter Matrix Elements
Enter decimal numbers. Empty cells are treated as 0.
Null Space Results
Rank-Nullity Theorem: Rank + Nullity = Number of Columns
Verification: 2 + 1 = 3 = 3 ✓
Basis for Null Space
Vector 1:
Quick Examples
Key Concepts
Null Space
Set of all vectors v where Av = 0
Rank
Number of linearly independent rows/columns
Nullity
Dimension of the null space
Solution Steps
Perform Gauss-Jordan elimination
Identify leading ones (pivot columns)
Find free variables (non-pivot columns)
Construct basis vectors for null space
Verify using rank-nullity theorem
Understanding Null Space
What is the Null Space?
The null space (or kernel) of a matrix A is the set of all vectors v such that Av = 0, where 0 is the zero vector. It represents the "directions" that are completely eliminated by the linear transformation represented by matrix A.
Why is it Important?
- •Solving homogeneous systems of equations
- •Understanding linear transformations
- •Determining matrix properties and invertibility
- •Applications in physics and engineering
Key Theorems
Rank + Nullity = Number of Columns
(Rank-Nullity Theorem)
Properties
- Vector Space: Null space is always a vector subspace
- Zero Vector: Always contains the zero vector
- Linear Combinations: Closed under addition and scalar multiplication
- Basis: Can be described by a finite set of basis vectors
Note: If nullity = 0, the matrix has full column rank and is injective (one-to-one).