Null Space Calculator: Find the Nullity & Basis Instantly
Null Space Calculator
Calculate the null space (kernel), nullity, rank, RREF, and basis vectors of any matrix.
Problem Statement
Find the null space, nullity, and basis for matrix A:
| 1 | -1 | -1 |
| 2 | -2 | 1 |
Step 1: Reduced Row Echelon Form (RREF)
Matrix A par row operations apply karne ke baad:
| 1 | -1 | 0 |
| 0 | 0 | 1 |
Step 2: Solve Homogeneous System
Matrix equation: rref(A) × x = 0
| 1 | -1 | 0 |
| 0 | 0 | 1 |
| x1 |
| x2 |
| x3 |
| 0 |
| 0 |
Is se do equations milti hain:
- x1 – x2 = 0 → x1 = x2
- x3 = 0
Agar hum free variable x2 = t rakhein:
- x1 = t
- x2 = t
- x3 = 0
Step 3: Vector Form
| x1 |
| x2 |
| x3 |
| t |
| t |
| 0 |
| 1 |
| 1 |
| 0 |
Final Answer
Basis for Null Space: { [1, 1, 0]T }
Nullity: 1
FAQs
What is null space in a matrix?
The null space (or kernel) of a matrix is the set of all vectors that, when multiplied by that matrix, result in a vector of pure zeros. Think of it as a mathematical “black hole”—no matter what non-zero input vector you throw in from this specific collection, the matrix completely flattens it down to zero.
Is null space a real thing?
Yes, it is a core concept in linear algebra with very real applications in engineering, physics, and computer science. While you cannot physically hold a null space, it practically represents the hidden “blind spots,” equilibrium states, or redundant information within a system—such as identifying which forces in a structure cause zero total movement.
Is 0 always in the nullspace?
Yes, the zero vector is always in the null space of every matrix. Because multiplying any matrix by a vector of all zeros guaranteed gives zero (A0 = 0), zero is an automatic member—which is also why every null space forms a valid mathematical subspace.
Is Ax = 0 the null space?
The equation Ax = 0 is the condition that defines the null space, rather than the space itself. The null space is the actual collection of every vector x that successfully solves that equation.
Are null and zero matrix the same?
No, they are completely different concepts: a zero matrix is a specific grid filled entirely with zero entries, whereas a null space is a set of vector inputs that get crushed to zero when transformed by a matrix. Even a matrix full of non-zero numbers can have a well-defined null space.
How to calculate a null space?
To find the null space, set up the system Ax = 0, construct the augmented matrix [A | 0], and use Gaussian elimination to reduce it to Row Echelon Form. From there, identify the free variables (columns without pivot positions) and solve for the dependent variables in terms of those free variables to write the general vector solution.
How to find the base of a null space?
After solving Ax = 0 in terms of your free variables, factor each free variable out of the final vector solution so it sits as a multiplier. The distinct, linearly independent vectors attached to those free variables form the basis—the fundamental building blocks that span the entire null space.
How do you calculate the dimension of the null space of a matrix?
The dimension of the null space (known as the nullity) is simply the total number of free variables in the matrix after reducing it to row echelon form. You can also calculate it instantly using the Rank-Nullity Theorem: take the total number of columns in the matrix and subtract the matrix’s rank (the number of pivot columns).
What happens if Wronskian is 0?
If the Wronskian determinant of a set of functions equals zero everywhere on an interval, it strongly suggests that the functions are linearly dependent—meaning at least one function can be written as a combination of the others. For solutions to a linear differential equation, a zero Wronskian guarantees they do not form a complete, independent set of solutions.
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