How To Find Basis Of Null Space

How to Find the Basis of a Null Space

Finding the basis of a null space is a fundamental concept in linear algebra, crucial for understanding concepts like linear transformations, rank-nullity theorem, and solving systems of linear equations. This article will guide you through the process, providing a step-by-step approach with examples to solidify your understanding. The null space, also known as the kernel, of a matrix represents all vectors that, when multiplied by the matrix, result in the zero vector. Finding its basis means finding a set of linearly independent vectors that span this null space.

What is a Null Space?

The null space of a matrix A, denoted as Null(A), is the set of all vectors x such that Ax = 0, where 0 is the zero vector. Understanding the null space provides insights into the matrix's properties and the solutions to the associated homogeneous system of linear equations.

Steps to Find the Basis of the Null Space

Let's break down the process into manageable steps using row reduction (Gaussian elimination):

1. Form the Augmented Matrix: Start by creating an augmented matrix [A | 0], where A is your given matrix and 0 is a column vector of zeros with the same number of rows as A.

2. Row Reduce to Reduced Row Echelon Form (RREF): Perform row operations (swapping rows, multiplying a row by a nonzero scalar, adding a multiple of one row to another) to transform the augmented matrix into its reduced row echelon form. This is a systematic way to solve the system of linear equations Ax = 0.

3. Identify Free and Pivot Variables: In the RREF matrix, identify the pivot columns (columns containing leading 1s) and the free columns (columns without leading 1s). The variables corresponding to the free columns are called free variables, while those corresponding to pivot columns are pivot variables.

4. Express Pivot Variables in Terms of Free Variables: Solve the system of equations represented by the RREF matrix for the pivot variables in terms of the free variables. This means expressing each pivot variable as a linear combination of the free variables.

5. Write the General Solution as a Linear Combination: The general solution will be a vector where the pivot variables are expressed as functions of the free variables, and the free variables act as parameters.

6. Extract Basis Vectors: The coefficients of the free variables in the general solution form the basis vectors for the null space. Each free variable corresponds to a basis vector.

Example:

Let's find the basis of the null space for the matrix:

A = ``` [[1, 2, -1], [2, 4, -2]]

1. **Augmented Matrix:**  [A | **0**] = ```
[[1, 2, -1 | 0],
 [2, 4, -2 | 0]]

2. Row Reduction: Performing row operations (R2 - 2R1 -> R2), we get: ``` [[1, 2, -1 | 0], [0, 0, 0 | 0]]

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

4. **Express Pivot Variable:** From the RREF, we have x₁ + 2x₂ - x₃ = 0, which means x₁ = -2x₂ + x₃.

5. **General Solution:** The general solution is:

**x** = ```
[[-2x₂,
  x₂,
  x₃]]
``` = x₂[-2, 1, 0] + x₃[1, 0, 1]

6. **Basis Vectors:** The basis vectors for the null space of A are [-2, 1, 0] and [1, 0, 1].


**Conclusion:**

Finding the basis of a null space is a powerful tool in linear algebra. Mastering this process allows for a deeper understanding of matrix properties and the solutions to systems of linear equations. By following these steps and practicing with various examples, you will develop proficiency in this essential technique.  Remember to always check your work – ensuring your basis vectors are linearly independent and span the entire null space is crucial.