Null Space Calculator
Matrix (one row per line)
The null space (kernel) of a matrix A is the set of all vectors x such that Ax = 0. The nullity is the dimension of the null space. Understanding null space is essential in solving linear systems and linear algebra.
- 1Solve Ax = 0 using row reduction (RREF)
- 2Free variables correspond to null space dimensions
- 3Rank-nullity theorem: rank + nullity = n (columns)
- 4Null space is always a subspace containing the zero vector
[[1,2,3],[4,5,6]] × x = 0=Null space has dimension 1; one free variable
| Matrix | Rank | Nullity | Free variables |
|---|---|---|---|
| 3×3 full rank | 3 | 0 | 0 |
| 3×3 rank 2 | 2 | 1 | 1 |
| 2×4 rank 2 | 2 | 2 | 2 |
| Identity n×n | n | 0 | 0 |
References
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