How To Find Basis For Null Space

How to Find a Basis for the Null Space of a Matrix

Finding the basis for the null space of a matrix is a fundamental concept in linear algebra with significant applications in various fields, including computer science, engineering, and statistics. The null space, also known as the kernel, represents all vectors that, when multiplied by the matrix, result in the zero vector. This article will guide you through the process of finding this basis, explaining the underlying concepts and providing a step-by-step approach. Understanding this process is crucial for solving systems of linear equations and comprehending the properties of linear transformations.

What is the Null Space?

The null space of an m x n matrix A, denoted as Null(A), is the set of all vectors x in Rⁿ such that Ax = 0, where 0 is the zero vector. In simpler terms, it's the set of all solutions to the homogeneous system of linear equations Ax = 0. The null space is always a subspace of Rⁿ.

Finding the Basis: A Step-by-Step Guide

The key to finding a basis for the null space lies in solving the homogeneous system Ax = 0. This involves using techniques like Gaussian elimination or row reduction to transform the augmented matrix [A | 0] into its reduced row echelon form (RREF). Let's break down the process:

1. Row Reduction to Reduced Row Echelon Form (RREF):

This is the crucial first step. Use Gaussian elimination or other row reduction methods to transform matrix A into its RREF. Remember that performing row operations on A does not change the solution set of Ax = 0.

2. Identifying Free and Pivot Variables:

Once in RREF, identify the pivot columns (columns with leading 1's) and the free columns (columns without leading 1's). The number of free variables will equal the dimension of the null space (also known as the nullity). Pivot variables correspond to leading entries in the RREF, while free variables can take on any value.

3. Expressing Pivot Variables in Terms of Free Variables:

From the RREF, express each pivot variable in terms of the free variables. This will give you a general solution to Ax = 0.

4. Constructing the Basis Vectors:

For each free variable, assign it a value of 1 while setting all other free variables to 0. Then, calculate the corresponding values of the pivot variables using the expressions derived in step 3. This process generates a vector that is a solution to Ax = 0. Repeat this process for each free variable. The set of vectors you obtain forms a basis for the null space.

Example:

Let's consider the matrix:

A =  [ 1  2  3 ]
     [ 4  5  6 ]
     [ 7  8  9 ]

1. Row Reduction: Performing row reduction on A (you can use online calculators or perform this manually) yields the RREF:

RREF(A) = [ 1  0  -1 ]
           [ 0  1   2 ]
           [ 0  0   0 ]

2. Variables: The pivot variables are x₁ and x₂, and the free variable is x₃.

3. Expressing Pivot Variables: From the RREF, we get:

• x₁ = x₃
• x₂ = -2x₃

4. Basis Vectors: Setting x₃ = 1, we get x₁ = 1 and x₂ = -2. Therefore, one basis vector is:

v₁ = [ 1 ]
     [ -2 ]
     [ 1 ]

The null space of A has dimension 1 (one free variable), and its basis is {v₁}.

Conclusion:

Finding the basis for the null space of a matrix is a systematic process involving row reduction, variable identification, and constructing basis vectors based on free variables. Understanding this process is essential for mastering linear algebra and its various applications. Remember to practice with different matrices to solidify your understanding. This technique is fundamental for solving linear systems and understanding the characteristics of linear transformations, making it a valuable tool in various mathematical and computational contexts.