Find basis for left null space (y^T A = 0) via RREF of A^T
Enter 2x2 Matrix
Enter 3x3 Matrix
Enter 3x2 Matrix
Result
Step-by-Step Derivation
Left Null Space Formula
Left Null(A) = { y | y^T A = 0 }
Left Null(A) = Null(A^T)
dim(Left Null) = m - rank(A)
Compute RREF(A^T) to find basis
The left null space of matrix A consists of all vectors y such that y^T A = 0. It equals the null space of A^T and is found by reducing A^T to RREF.
⚠Left null space lives in ℝᵐ where m is the number of rows of A. For full row rank matrices, left null space contains only the zero vector.
What is Left Null Space?
The left null space (or left kernel) of an m×n matrix A is the set of all m-dimensional vectors y such that y^T A = 0. It represents the linear dependencies among the rows of A.
Transpose
Left null space of A = null space of A^T. Compute RREF of transpose.
Left Nullity
Dimension = m - rank(A). Measures row dependencies.
Row Dependencies
Basis vectors represent linear dependencies among rows of A.
Orthogonality
Left null space is orthogonal complement of column space of A.
Teaching Example: A = [[1,2],[2,4],[3,6]]
1. A^T = [[1,2,3],[2,4,6]]
2. RREF(A^T) = [[1,2,3],[0,0,0]]
3. Free columns: 2, 3
4. Left null space basis: [-2,1,0]^T, [-3,0,1]^T
5. Left nullity = 2
Applications
Linear SystemsRow DependenciesOrthogonal ProjectionControl TheoryMachine Learning
Frequently Asked Questions
What is left null space?▼
Left null space of A is set of vectors y such that y^T A = 0. It equals null space of A^T.
How to find left null space?▼
1. Compute RREF of A^T. 2. Identify free variables. 3. Solve y^T A = 0 to get basis vectors.
Left null space vs null space?▼
Null space solves Ax=0 (right). Left null space solves y^T A=0. N(A^T) = left null space.
What is left nullity?▼
Left nullity = dim(left null space) = m - rank(A), where m is number of rows.
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