How To Find The Null Space

How to Find the Null Space of a Matrix

Finding the null space of a matrix is a fundamental concept in linear algebra with applications in various fields, including computer graphics, machine learning, and cryptography. This article will guide you through the process, explaining the concepts and providing a step-by-step approach. Understanding the null space involves grasping the idea of linear transformations, vector spaces, and solving systems of linear equations.

The null space (also known as the kernel) of a matrix A, denoted as Null(A), is the set of all vectors x such that Ax = 0. In simpler terms, it's the set of all vectors that, when multiplied by the matrix A, result in the zero vector. The dimension of the null space is called the nullity of the matrix.

Understanding the Process

Finding the null space involves solving a homogeneous system of linear equations. This means we're looking for solutions to Ax = 0, where 0 is the zero vector. The key to solving this is using Gaussian elimination (row reduction) to find the reduced row echelon form (RREF) of the matrix.

Step-by-Step Guide

Let's illustrate the process with an example. Consider the matrix:

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

Step 1: Form the Augmented Matrix

First, we form the augmented matrix [A | 0], where 0 is the zero vector with the same number of rows as A. In our example:

[A | 0] = [ 1  2  3 | 0 ]
          [ 4  5  6 | 0 ]
          [ 7  8  9 | 0 ]

Step 2: Perform Gaussian Elimination

Now, we perform Gaussian elimination (row reduction) to transform the augmented matrix into its reduced row echelon form (RREF). The goal is to obtain a matrix where:

• Leading entries (the first non-zero element in each row) are 1.
• Leading entries are the only non-zero elements in their columns.
• Rows with all zeros are at the bottom.

The specific row operations used (swapping rows, multiplying rows by constants, adding multiples of one row to another) will depend on the matrix. After performing Gaussian elimination on our example, we might obtain (the exact RREF depends on the specific row operations chosen):

RREF = [ 1  0 -1 | 0 ]
       [ 0  1  2 | 0 ]
       [ 0  0  0 | 0 ]

Step 3: Express Free Variables in Terms of Parameters

Notice that the third column doesn't have a leading 1. This means the corresponding variable (let's call it x₃) is a free variable. We can assign it a parameter, say x₃ = t, where t is any scalar.

Step 4: Express Basic Variables in Terms of Free Variables

Now, we express the basic variables (those corresponding to columns with leading 1s) in terms of the free variables. From the RREF, we have:

• x₁ - x₃ = 0 => x₁ = x₃ = t
• x₂ + 2x₃ = 0 => x₂ = -2x₃ = -2t

Step 5: Write the Null Space as a Linear Combination

Finally, we can express the solution vector x as a linear combination of the free variable parameter:

x = [ x₁ ]   = [ t ]
    [ x₂ ]   = [ -2t ]
    [ x₃ ]   = [ t ]
    = t [ 1 ]
      [ -2 ]
      [ 1 ]

This shows that the null space of A is spanned by the vector [1, -2, 1]ᵀ. Any scalar multiple of this vector will also be in the null space. Therefore, the null space is the line through the origin defined by this vector.

Higher Dimensional Cases

The process remains the same for larger matrices. You'll simply have more variables and potentially more free variables. The number of free variables equals the dimension of the null space (nullity). The vectors corresponding to the free variables form a basis for the null space.

Understanding and correctly applying these steps is crucial for mastering linear algebra and its various applications. Remember that practice is key to solidifying your understanding of finding the null space of a matrix.