Find basis for null space (kernel) via RREF and free variables
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Result
Step-by-Step Derivation
Null Space Formula
N(A) = { x | Ax = 0 }
dim(N(A)) = nullity = n - rank(A)
Find RREF(A), identify pivot/free columns
Set free variables = 1, solve for pivot vars
The null space (kernel) of matrix A consists of all vectors x that satisfy Ax = 0. It's found by reducing A to row-reduced echelon form (RREF), identifying free variables, and solving for pivot variables in terms of free variables.
⚠If rank(A) = n (full column rank), null space contains only the zero vector. The nullity equals the number of free variables in RREF.
What is Null Space?
The null space (or kernel) of an m×n matrix A is the set of all n-dimensional vectors x such that Ax = 0. Geometrically, it represents the solution space of a homogeneous linear system. The dimension of the null space is called the nullity.
Pivot Columns
Columns with leading 1s in RREF. Corresponding variables are pivot variables determined by free variables.
Free Variables
Columns without leading 1s. Can take any value, used to parameterize null space basis.
Nullity
Dimension of null space = number of free variables = n - rank(A).
RREF
Row-reduced echelon form simplifies finding null space by exposing pivot/free structure.
Teaching Example: A = [[1,2],[2,4]]
1. RREF(A) = [[1,2],[0,0]]
2. Pivot column: 1, Free column: 2
3. Let x2 = t (free variable)
4. From RREF: x1 = -2t
5. Null space basis: [-2, 1]^T
Applications
Linear SystemsLeast SquaresControl TheoryImage ProcessingMachine Learning
Frequently Asked Questions
What is null space?▼
Null space (kernel) of matrix A is the set of all vectors x such that Ax = 0. It represents all solutions to the homogeneous equation.
How to find null space basis?▼
1. Compute RREF of A. 2. Identify free variables. 3. Set each free variable to 1, others to 0. 4. Solve for pivot variables.
What is nullity?▼
Nullity is dimension of null space = number of columns - rank(A). It counts free variables in RREF.
Null space vs column space?▼
Null space contains solutions to Ax=0. Column space contains all possible Ax. They are orthogonal complements.
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