1. What is Null Space in Linear Algebra?

Definition: The null space (or kernel) of a matrix A is the set of all vectors \(\vec{x}\) that satisfy the equation \(A\vec{x} = \vec{0}\).

Purpose: Finding the null space helps understand the solutions to homogeneous systems of linear equations and reveals linear dependencies in the matrix.

2. How Does the Null Space Calculator Work?

The calculator solves the equation:

Where:

• \( A \) — Input matrix (m×n dimensions)
• \( \vec{x} \) — Solution vector (n×1 dimensions)
• \( \vec{0} \) — Zero vector (m×1 dimensions)

Explanation: The calculator performs row reduction to find the basis vectors that span the null space of matrix A.

3. Importance of Null Space Calculation

Details: Understanding the null space is crucial for solving linear systems, analyzing linear transformations, and studying matrix properties like rank and invertibility.

4. Using the Calculator

Tips: Enter your matrix using commas to separate columns and semicolons to separate rows. All entries should be numeric values.

5. Frequently Asked Questions (FAQ)

Q1: What does a non-trivial null space indicate?
A: A non-trivial null space (with more than just the zero vector) means the matrix has linearly dependent columns and is not full rank.

Q2: How is null space related to solutions of Ax=b?
A: The null space describes all solutions to the homogeneous equation Ax=0. For Ax=b, solutions are particular solution + null space vectors.

Q3: What's the dimension of the null space called?
A: It's called the nullity of the matrix, which equals the number of free variables in the system.

Q4: Can a full-rank matrix have a null space?
A: Only the trivial null space (zero vector). A full-rank square matrix has nullity zero.

Q5: How do I interpret the basis vectors?
A: Each basis vector represents an independent solution to Ax=0. Any linear combination of these vectors is also in the null space.