## Introduction to Null Space Calculator:

Null space calculator is an online tool that helps you to **find the null space** of a given matrix in a few seconds. Our tool evaluates different orders of the matrix, whether it is 2 by 2 order or 3 by 4 matrix, which makes the process of null space matrix to get a solution.

The null space of a matrix calculator is a beneficial tool for students or professionals because when you deal with a higher-order matrix, you may get confused while solving its solution. Our tool is specially designed to provide accurate solutions to null space problems.

## What is Null Space?

The null space of a square matrix is a fundamental concept in linear algebra. It refers to the set of all vectors x in the domain of A that **span the zero vector** under the linear transformation in vector space. It is also known as the kernel of a given matrix.

The null space Null(A) can be denoted as:

$$ Null\;(A) \;=\; \{x∈R^n ∣ Ax=0 \} $$

Where:

- A is a m×n matrix, A vector map from R
^{n}(the domain) to R^{m}(the codomain). - x is a vector in R
^{n}. - 0 is the zero vector in R
^{m}.

The null space provides insight into the behavior of linear transformations in vector spaces.

## How to Calculate the Null Space of a Matrix?

For the **calculation of the null space** (kernel) of a matrix, the null space calculator finds all vectors x such that Ax=0, as A is the given matrix. Let us see how you compute the null space matrix with the help of an example.

### Example: Find the Null Space of the following Matrix:

$$ \left[ \begin{matrix} 2 & 4 & 3 \\ 6 & 3 & 1 \\ 2 & 5 & 0 \\ \end{matrix} \right] $$

**Solution for step-by-step calculation:**

- Combine the matrix A with the zero vector on the right-hand side to form an augmented matrix [Ax∣0]. $$ \left[ \begin{matrix} 2 & 4 & 3 \\ 6 & 3 & 1 \\ 2 & 5 & 0 \\ \end{matrix} \right], \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right] $$

- Use the Gaussian elimination method on the augmented matrix to get a reduced row echelon form (RREF). $$ R_1 \leftarrow R_1 \div 2 $$ $$ =\; \left[ \begin{matrix} 1 & 2 & 1.5 \\ 6 & 3 & 1 \\ 2 & 5 & 0 \\ \end{matrix} \right] $$ $$ R_2 \leftarrow R_2 - 6 \times R_1 $$ $$ =\; \left[ \begin{matrix} 1 & 2 & 1.5 \\ 0 & -9 & -8 \\ 2 & 5 & 0 \\ \end{matrix} \right] $$

$$ =\; \left[ \begin{matrix} 1 & 2 & 1.5 \\ 0 & -9 & -8 \\ 0 & 1 & -3 \\ \end{matrix} \right] $$

$$ R_2 \leftarrow R_2 \div -9 $$

$$ =\; \left[ \begin{matrix} 1 & 2 & 1.5 \\ 0 & 1 & 0.88888889 \\ 0 & 1 & -3 \\ \end{matrix} \right] $$

$$ R_1 \leftarrow R_1 - 2 \times R_2 $$

$$ =\; \left[ \begin{matrix} 1 & 0 & -0.27777778 \\ 0 & 1 & 0.88888889 \\ 0 & 0 & -3.88888889 \\ \end{matrix} \right] $$

$$ R_3 \leftarrow R_3 - R_2 $$

$$ =\; \left[ \begin{matrix} 1 & 0 & -0.27777778 \\ 0 & 1 & 0.88888889 \\ 0 & 0 & -3.88888889 \\ \end{matrix} \right] $$

$$ R_3 \leftarrow R_3 \times -0.25714286 $$

$$ =\; \left[ \begin{matrix} 1 & 0 & -0.277777778 \\ 0 & 1 & 0.88888889 \\ 0 & 0 & 1 \\ \end{matrix} \right] $$

$$ R_1 \leftarrow R_1 + 0.27777778 \times R_3 $$

$$ =\; \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0.88888889 \\ 0 & 0 & 1 \\ \end{matrix} \right] $$

$$ R_2 \leftarrow R_2 - 0.88888889 \times R_3 $$

$$ =\; \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] $$

- After getting the reduced echelon form using gauss elimination method, write the basis vectors of the null space such as,

Now solve the matrix equation,

$$ \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right] \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right] $$

$$ x_1 \;=\; 0 $$

$$ x_2 \;=\; 0 $$

$$ x_3 \;=\; 0 $$

- Write the null space as the span of the basis vectors that get after finding the value of variables.

Add equation for each free variable,

$$ x_1 \;=\; 0 $$

$$ x_2 \;=\; 0 $$

$$ x_3 \;=\; 0 $$

Solve for each variable in terms of the free variables,

$$ x_1 \;=\; 0 $$

$$ x_2 \;=\; 0 $$

$$ x_3 \;=\; 0 $$

Convert this into vectors,

$$ \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right] $$

If you need to solve null space problems quickly and accurately, try our null space calculator. It's advanced, easy to use, and completely free.

## How to Use a Null Space of a Matrix Calculator?

The nullity calculator has a simple layout that helps everyone easily get the solutions to a given null space matrix problem. You just need to put your problem in this matrix nullity calculator and immediately get the result. These steps are:

- Add the size of the matrix in the input field.
- Enter the elements of the matrix in the input field.
- Check your input matrix values before clicking on the calculate button to get the result of the null space question.
- Click the “
**Calculate**” button to solve the null space problems. - Click the “Recalculate” button to evaluate more examples of the null space calculator working method used to get the solution.

## Outcome of Null Space Basis Calculator:

Matrix kernel calculator provides you with the **solution of null space** questions when you click on the calculate button. It may include as:

**In the Result Box**

When you click on the result button you get the solution of the null space matrix using the reduced echelon form

**Possible Steps Box**

Click on the steps option so that you get the solution to the null space matrix in steps.

## Advantages of Using Nullity Calculator:

It has millions of advantages that you avail when you use it to calculate the given null space problem with a solution. These advantages are:

- A matrix nullity calculator is a
**trustworthy tool**that always provides you with an accurate solution for various types of null space matrix problems. - It is a swift tool that helps you evaluate null space problems with solutions in a few seconds without any difficulty.
- Our matrix null space calculator is a learning tool that helps you to enhance your knowledge about null space in linear algebra without going to any teacher.
- It is a manageable tool that can solve null space problems of different order without taking any external effort.
- Null space basis calculator is a free tool that allows you to use it for the calculation of null space problems.
- The kernel of matrix calculator can handle null space problems of varying complexities and matrix sizes, making it adaptable to different academic or professional needs without requiring additional tools or resources.