# Matrix Null Space (Kernel) and Nullity Calculator

## Find null spaces step by step

The calculator will find the null space (kernel) and the nullity of the given matrix, with steps shown.

The Null Space Calculator is your reliable ally in finding the null space of a matrix. Created to swiftly and accurately find the null space of any matrix, this tool seamlessly merges cutting-edge technology with an intuitive interface.

## How to Use the Matrix Null Space and Nullity Calculator?

### Input

Enter the elements of your matrix into the specified fields of the calculator.

### Calculation

Once your matrix is correctly entered, click on the "Calculate" button.

### Result

The calculator will promptly process your request and present the null space of your matrix as the set of basis vectors.

## What Is Null Space?

In linear algebra, the null space (often referred to as the kernel) comprises all vectors that give the zero vector when multiplied by a particular matrix.

For example, consider a matrix $$$A$$$ and a vector $$$\mathbf{\vec{x}}$$$. If multiplying $$$A$$$ by $$$\mathbf{\vec{x}}$$$ yields the zero vector, then $$$\mathbf{\vec{x}}$$$ is a part of the null space of the matrix $$$A$$$.

In formula terms, if $$$A$$$ is some matrix and there is a vector $$$\mathbf{\vec{x}}$$$ such that $$$A\mathbf{\vec{x}}=0$$$, then $$$\mathbf{\vec{x}}$$$ is in the null space of $$$A$$$.

These vectors in the null space offer valuable information about the matrix's properties and play a crucial role in solving systems of linear equations.

## What Is Nullity?

Nullity, in the context of linear algebra, is the number of basis vectors in the null space of a given matrix.

To illustrate, suppose we have a matrix whose null space can be spanned by three linearly independent vectors. In this case, the nullity of that matrix is three.

Nullity is a very important property of a matrix; it provides significant insights about the matrix's properties, particularly, its rank and invertibility. This is underlined by the rank-nullity theorem, which establishes that the sum of the rank and the nullity of a matrix equals the total number of its columns.

In a formulaic representation, if $$$r$$$ is the rank of the matrix, $$$n$$$ is the nullity and $$$c$$$ is the total number of columns, then the rank-nullity theorem states that $$$r+n=c$$$.

## Why Choose Our Matrix Null Space and Nullity Calculator?

### Efficiency

Our calculator swiftly computes the null space and nullity of a given matrix, saving you from lengthy manual calculations.

### Precision

Leveraging advanced mathematical algorithms, the calculator ensures accurate results every time.

### User-Friendly Interface

With its simple and intuitive interface, the tool is easy to use for both beginners and experts in linear algebra.

### Versatility

The calculator does not only compute the null space and nullity, but it also provides step-by-step solutions.

### Accessibility

As an online tool, it's accessible anytime and anywhere you need it. Whether you're studying late at night or crunching numbers at the office, our calculator is just a few clicks away.

### FAQ

#### How Is the nullity of a matrix related to its rank?

The rank and nullity of a matrix are interconnected through the rank-nullity theorem, which states that the sum of a matrix's rank and its nullity equals the total number of its columns.

#### Can I use this calculator to compute the rank of a matrix?

No, there is another calculator for calculating the rank of a matrix. If you want to use this calculator, the rank of a matrix equals the number of columns of the matrix minus the number of vectors in its null space.

#### What is the null space of a matrix?

The null space of a matrix, also known as the kernel, comprises all vectors that, when multiplied by the matrix, yields the zero vector. In essence, if $$$A$$$ is a matrix, then the null space of $$$A$$$ consists of all vectors $$$\mathbf{\vec{x}}$$$ such that $$$A\mathbf{\vec{x}}=0$$$.

#### How do I use the Null Space Calculator?

Using the Null Space Calculator is simple. First, input the elements of the matrix into the calculator. Once done, click the "Calculate" button and the calculator will provide the null space of your matrix as the set of basis vectors.