Dimension of the null space (number of free variables)
Rank2
Null space basisBasis: { [1, -2, 1] }
RREF[1 0 -1]
[0 1 2]
[0 0 0]
Parametric solutionx1 = t
x2 = -2 * t
x3 = t (free)
Rank2
Nullity1
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Null space has dimension 1 (rank = 2, nullity = 1).
Rank-nullity check: rank (2) + nullity (1) = 3, which equals the number of columns (3). The theorem is satisfied.
The null space is 1-dimensional. There are 1 free variable, meaning infinitely many solutions to A x = 0.
Each basis vector shown represents one direction in which x can move while keeping A x = 0. Any linear combination of those vectors is also in the null space.
The matrix is rank-deficient (rank is less than the smaller dimension), so the rows are not all linearly independent.
Next stepTo solve a non-homogeneous system A x = b, find one particular solution and add the general null space solution.