1. What Is Null Space?

The null space of a matrix A is the set of all vectors x that satisfy the equation A x = 0. It represents the solution space of the homogeneous system of linear equations and provides insight into the linear dependence of the matrix columns.

2. How The Calculator Works

The calculator uses Gaussian elimination (row reduction) to transform the matrix to reduced row echelon form. The basis vectors of the null space are then extracted from the free variables in the solution.

Algorithm Steps:

1. Create augmented matrix [A | I] where I is identity matrix
2. Perform row operations to achieve reduced row echelon form
3. Identify free variables (columns without pivots)
4. Extract basis vectors from the corresponding columns of the identity part

3. Applications Of Null Space

Details: Null space calculations are fundamental in linear algebra with applications in solving systems of equations, understanding linear transformations, computer graphics, data analysis, and engineering problems involving constraint satisfaction.

4. Using The Calculator

Tips: Enter the matrix with rows separated by newlines and elements within each row separated by commas. The calculator will automatically compute the basis vectors for the null space using row reduction techniques.

5. Frequently Asked Questions (FAQ)

Q1: What does it mean if the null space is trivial?
A: If the null space contains only the zero vector, the matrix has full column rank and its columns are linearly independent.

Q2: How is null space related to column space?
A: The null space and column space are orthogonal complements. The dimension of null space plus the dimension of column space equals the number of columns.

Q3: Can a matrix have an empty null space?
A: No, every matrix has at least the zero vector in its null space. The null space is never empty.

Q4: What is the difference between null space and kernel?
A: Null space typically refers to matrices while kernel refers to linear transformations, but they represent the same concept - the set of vectors mapped to zero.

Q5: How does row reduction help find the null space?
A: Row reduction preserves the solution space of A x = 0 while making it easier to identify free variables and extract basis vectors.