The null space (also called the kernel) of a matrix A is the set of all vectors x such that when multiplied by A, the result is the zero vector. Mathematically, it is defined as all solutions x to the equation Ax = 0. The null space reveals crucial information about a matrix, including whether the system has a unique, infinite, or no solution.
Understanding the null space is essential for exploring the linear independence, dimension, and rank of matrices in various theoretical and applied domains.
Detailed Explanation of the Calculator’s Working
The Null Space of a Matrix Calculator works by solving the matrix equation Ax = 0 using Gaussian elimination or row-reduction to reduced row echelon form (RREF). Once the matrix is simplified, the calculator identifies the free variables that describe the solution space. These variables are used to generate a basis for the null space, effectively giving the general solution.
This tool supports matrices of any size and works with real numbers or rational inputs. The output includes step-by-step row operations, the reduced form of the matrix, and the null space expressed as a span of basis vectors.
Formula with Variables Description
Null(A) = { x | Ax = 0 }
Where:
A = m × n matrix
x = n × 1 column vector (variable vector)
Ax = matrix multiplication of A and x
0 = zero vector of dimension m × 1
This formula represents the set of all vectors x that satisfy the condition Ax = 0.
Quick Reference Table: Common Linear Algebra Terms
| Term | Description |
|---|---|
| Null Space | Set of all vectors x where Ax = 0 |
| Rank | Number of linearly independent rows or columns |
| RREF | Reduced Row Echelon Form |
| Free Variable | Variable that can take any value in a solution to a linear system |
| Linearly Dependent | Vectors that can be written as a linear combination of others |
| Homogeneous System | A system of linear equations where all constant terms are zero |
| Basis Vectors | Set of vectors that span the null space |
| Dimension of Null(A) | Also known as the nullity of matrix A |
Example
Let’s consider the matrix A:
A = [ 2 4 6 ]
[ 1 3 5 ]
To find the null space:
- Set up the matrix equation Ax = 0
- Apply row reduction to get RREF: cssCopyEdit
[ 1 0 -1 ] [ 0 1 2 ] - Express the solution in terms of a free variable, say t
- The null space is: plaintextCopyEdit
x = t * [-1, -2, 1]ᵗ
Thus, the null space is spanned by the vector [-1, -2, 1], and every solution is a scalar multiple of this vector.
Applications
Linear Algebra Systems
In mathematics, null space is used to assess the solvability and structure of systems of linear equations. It determines if a system has infinitely many solutions and helps identify dependent variables.
Computer Graphics
Null spaces play an important role in graphics transformations, especially when detecting degenerate transformations (e.g., collapsing shapes into lower dimensions).
Engineering Simulations
Structural and mechanical engineers use null space concepts to simulate redundant constraints, analyze rigid body motions, or simplify finite element models in simulation software.
Most Common FAQs
The null space reveals the set of all vectors that, when multiplied by the matrix, result in a zero vector. This space gives insight into the matrix's structure, particularly how many free variables exist in a system of linear equations. A non-trivial null space indicates linear dependence among columns.
Yes. When a matrix has full column rank, its null space contains only the zero vector. This means that all columns are linearly independent, and the system Ax = 0 has a unique solution: x = 0.
According to the Rank-Nullity Theorem, the dimension of the null space plus the rank of the matrix equals the number of columns. This theorem connects the structural properties of a matrix to its solution spaces.