1. What is Matrix Null Space?

The null space of a matrix A, denoted N(A), is the set of all vectors x such that Ax = 0. It represents all solutions to the homogeneous system of linear equations defined by the matrix.

2. How Does the Calculator Work?

The calculator uses Gaussian elimination to:

Steps:

• Convert matrix to row echelon form
• Identify pivot and free variables
• Express basis vectors in terms of free variables

3. Importance of Null Space Calculation

Applications: Understanding solutions to homogeneous systems, analyzing linear transformations, solving differential equations, and more.

4. Using the Calculator

Instructions: Enter matrix elements separated by spaces, with rows separated by semicolons. For example: "1 2 3; 4 5 6; 7 8 9".

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between null space and column space?
A: Null space contains solutions to Ax=0, while column space contains all possible outputs Ax.

Q2: When does a matrix have a non-trivial null space?
A: When the matrix has linearly dependent columns (rank less than number of columns).

Q3: How is null space related to linear independence?
A: A matrix has only the trivial null space (x=0) if and only if its columns are linearly independent.

Q4: What does the dimension of the null space represent?
A: The nullity of the matrix, which equals the number of free variables in the system.

Q5: Can the null space be empty?
A: No, it always contains at least the zero vector.