Basis For Null Space Calculator

Null Space Basis:

\[ N(A) = \{ x \in \mathbb{R}^n \mid Ax = 0 \} \]

Basis for Null Space:

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1. What is the Null Space?

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. A basis for the null space is a set of linearly independent vectors that span this space.

2. How Does the Calculator Work?

The calculator works by:

Converting the input matrix to reduced row echelon form (RREF)
Identifying pivot and free variables
Constructing basis vectors by setting each free variable to 1 (others to 0)

\[ \text{For } A \in \mathbb{R}^{m \times n}, \text{ solve } Ax = 0 \]

3. Importance of Null Space

Applications: The null space is fundamental in linear algebra for understanding solutions to homogeneous systems, linear transformations, and matrix properties.

4. Using the Calculator

Instructions: Enter your matrix with each row on a new line and elements separated by spaces. The calculator will display a basis for the null space.

5. Frequently Asked Questions (FAQ)

Q1: What does the dimension of the null space represent?
A: The dimension (nullity) equals the number of free variables in the RREF, indicating how many basis vectors are needed.

Q2: How is this related to the rank of the matrix?
A: Rank-Nullity Theorem: rank(A) + nullity(A) = number of columns in A.

Q3: Can the null space be empty?
A: No, it always contains at least the zero vector. The trivial null space {0} occurs when the only solution is x = 0.

Q4: What if my matrix has complex numbers?
A: This calculator currently handles real matrices only. Complex matrices would require extending the field to complex numbers.

Q5: How accurate is the calculation?
A: Results are numerically accurate for well-conditioned matrices. Ill-conditioned matrices may show precision limitations.