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Matrix Null Space Kernel Calculator + Online Solver With Free Steps

A Matrix Null Space Kernel Calculator is used to find the Null Space for any Matrix. The Null Space of a Matrix is a very important quantity as it corresponds to the quantities of the vectors concerning zeros.

The Null Space of a Matrix is therefore a description of the Subspace of the Euclidean Space the matrix tends to associate with. The Matrix Null Space Kernel Calculator thus works by solving the matrix against a zero-vector output.

What Is a Matrix Null Space Kernel Calculator?

A Matrix Null Space Kernel Calculator is an online calculator which is designed to solve your Null Space problems.

To solve a Null Space problem, a lot of computation is required, and that is why this calculator comes in very handy because it solves your problems in your browser without any requirements for downloads or installations.

Now, as any problem would go, you would require an initial input to solve. So is the requirement with the Matrix Null Space Kernel Calculator, as it requires a matrix as input. The Matrix is entered into the input box as a set of vectors, and then the rest is done by the calculator.

How To Use a Matrix Null Space Kernel Calculator?

To use a Matrix Null Space Kernel Calculator, you must first have a matrix as input for which you would like to find out the Null Space. And then, you would enter its entries into the input box, and at the press of a button, the calculator will solve your problem for you.

So, to get the best results from your Matrix Null Space Kernel Calculator, you may follow the given steps:

Step 1

You can start off by simply setting your problem into the right format. A matrix is 2-dimensional array, and it can be difficult to enter such set of data into a line. The method used for formatting is taking each row as a vector, and making a set of vectors such as:

\[A = \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix} = \{\{a, b, c\}, \{d, e, f\}, \{g, h, i\}\}\]

Step 2

Once, you have your matrix in the right format for the calculator, you can simply enter the set of vectors in the input box labeled as ker.

Step 3

Now, you don’t have to do anything other than just press the Submit button. And this will bring up the solution to your problem in a new interactable window.

Step 4

Finally, if you would like to solve any more questions of this sort, you can simply enter their inputs in the correct format in the opened interactable window.

An important fact to note about this calculator is that it will have trouble solving for Null Spaces of matrices with orders higher than 3 x 3 as the computation becomes very complex and lengthy moving up to the mark of 4 rows or columns.

How Does a Matrix Null Space Kernel Calculator Work?

A Matrix Null Space Kernel Calculator works by solving the Null Space for the provided matrix by using a long process where the input matrix is subjected to several different computations.

Therefore, in theory, it is mapping vectors to Zeros and then finding out their mathematical solutions for a given matrix A.

What Is a Matrix?

A Matrix is defined as a rectangular-shaped collection of numbers, quantities, symbols, etc. It is used very commonly in Mathematics and Engineering for storing and saving data.

A Matrix usually has a particular number of rows and columns set up in it. Plurally, a matrix is referred to as Matrices. They were initially used to solve systems of Linear Equations and have been used for this purpose for a long time until today. The oldest recorded use of simultaneous equations described using matrices was from the 2nd century BCE.

The entries or values inside the Matrix are referred to as cells or boxes. Therefore, a value in a particular row and column would be in that corresponding cell. There are so many different types of matrices which differ from one another based on their Order.

Types of Matrices

There are, therefore, so many different types of matrices. These matrices have unique orders associated with them. Now the most common one is the Row Matrix, a type of matrix which has only one row. This is a unique matrix as its order always remains of the form, 1 x x, whilst Column Matrices are the opposite of Row Matrices with only one column, and so on.

Null Matrix

A Null Matrix is s type of matrix which we’re going to use the most, it is also referred to as Zero Matrix. Thus, from a linear algebra standpoint, a null matrix corresponds to a matrix whose every entry is Zero.

Null Space or Kernel of a Matrix

We mentioned earlier that matrices are also known as Linear Maps in the dimensional analysis of space, whether it be 1, 2, 3, or even 4 D. Now, a Null Space for such a matrix is defined as the result of mapping vectors to a zero vector. This results in a subspace, and it is referred to as Null Space or Kernel of a Matrix.

Solve for Null Space

Now, let’s assume that we have a matrix of the form:

\[A = \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\]

Now, the Null Space solution for this would have to be given as:

Ax = 0

\[\begin{bmatrix} a & b & c \\ d & e & f  \\ g & h & i\end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}\]

Now, one more thing to take care of is solving the matrix A to simplification. This is done by using the Gauss-Jordan Elimination method, or also commonly known as Row-Reductions.

First, we clear out the left most column on the rows below:

\[\begin{bmatrix} a & b & c \\ d & e & f  \\ g & h & i\end{bmatrix} \rightarrow \begin{bmatrix} a & b & c \\ 0 & s & t  \\ 0 & v & w\end{bmatrix} \]

Then, we move further and clear both left columns on the 3rd row:

\[\begin{bmatrix}a & b & c \\ 0 & s & t  \\ 0 & v & w\end{bmatrix} \rightarrow \begin{bmatrix} a & b & c \\ 0 & s & t  \\ 0 & 0 & z\end{bmatrix} \]

And finally, we get the matrix in the Reduced Echelon form as follows:

\[\begin{bmatrix}a & b & c \\ 0 & s & t  \\ 0 & 0 & z\end{bmatrix} \rightarrow \begin{bmatrix} 1 & m & n \\ 0 & 1 & q  \\ 0 & 0 & 1\end{bmatrix} \]

Once simplified to something much more easily solvable i.e., Reduced Echelon form, we can simply solve for the Null Space of said matrix.

As this combination of matrices describe a system of linear equations:

\[\begin{bmatrix} 1 & m & n \\ 0 & 1 & q  \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}\]

We get these linear equations, the solution of which will give us the Null Space of the initial Matrix.

x1 + mx2 + nx3 = 0, x2 + qx3 = 0, x3 = 0

Properties of Null Space

There are a set of properties that are unique to the Null Space of a matrix, and they begin by exclaiming that, A . x = 0 has a “.” which represents matrix multiplication.

Moving forward, the properties of a Null Space are given below:

  1. A zero output for a matrix’s null space is always present in the Null Space. As for a Zero Vector, anything multiplied by it will result in a zero output.
  2. Another important property to note is that there can be as many as an infinite number of entries in the Null Space of a Matrix. And this depends on the Order of the Matrix in question.
  3. The last and most significant thing to know about a Null Space is that in the vector calculus of matrices, a kernel corresponds to a Subspace, and this subspace is a part of a bigger Euclidean Space.

Nullity of a Matrix

The nullity of a Matrix is a quantity that describes the dimensionality of the said matrix’s Null Space. It works hand in hand with the Rank of a Matrix.

So, if a matrix’s Rank corresponds to the Eigenvalues of a matrix that are non-zero, then Nullity tends towards those eigenvalues which are zero. To find the Nullity of a matrix, you can simply subtract from the number of columns of a matrix its Rank.

And both these quantities are found using the Gauss-Jordan Elimination method.

Solve for Nullity

Now, to solve for Nullity, you don’t require anything too far from what we have already been calculating. As in the solution for Null Space above, we found the Reduced Echelon form of a matrix. We will use that form to calculate the Rank and Nullity of the given matrix.

So let’s assume a matrix is reduced to this form:

\[\begin{bmatrix}a & b & c \\ d & e & f  \\ g & h & i\end{bmatrix} \rightarrow \begin{bmatrix} 1 & m & n \\ 0 & 1 & q  \\ 0 & 0 & 1\end{bmatrix} \]

Now, if we calculate the Rank of this Matrix, it comes out to be 3 as Rank describes the non-zero row number for any matrix in its Reduced Echelon Form. Now, given that this matrix has at least a 1 in each row, each row is a non-zero row.

Therefore, as the matrix is of Order: 3 x 3, we can solve this mathematical expression to find the Nullity for this matrix.

Number of Columns – Rank = Nullity

3 – 3 = 0

This generalized matrix can have a Nullity of 0.

Solved Examples

Example 1

Consider the following matrix:

\[A = \begin{bmatrix}2 & 1 \\ -4 & -2\end{bmatrix}\]

Find the Null Space for this matrix.

Solution

Let’s start by setting up our matrix input in the form of this equation, Ax = 0 given below:

\[Ax = \begin{bmatrix}2 & 1 \\ -4 & -2\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}\]

To solve for Null Space, you want to solve the Row-Reduced form for this matrix, also referred to as Reduced Echelon form using the Gauss-Jordan Elimination Method:

\[\begin{bmatrix}2 & 1 \\ -4 & -2\end{bmatrix} \rightarrow \begin{bmatrix}2 & 1 \\ 0 & 0\end{bmatrix}\]

Now, replacing the row-reduced matrix for the original gives us this result:

\[\begin{bmatrix}2 & 1 \\ 0 & 0\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}\]

Solving the first row gives us 2×1+x2 =0

And finally, we get the result of Null Space as:

\[\begin{bmatrix}x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}-x \\ 2x\end{bmatrix} : x \in \Re \]

Example 2

Determine the Null Space for the following matrix:

\[A = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}\]

Solution

Input the matrix in the form of this equation, Ax = 0 given as:

\[Ax = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}\]

Solve for the Null Space of the given matrix using the calculator.

Find the Row-Reduced form for this matrix, that is also referred to as Reduced Echelon form using the Gauss-Jordan Elimination Method.

\[\begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \rightarrow \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix} \rightarrow \begin{bmatrix}1 & 2 \\ 0 & -3\end{bmatrix}\]

Replacing the row-reduced matrix for the original gives us:

\[\begin{bmatrix}1 & 2 \\ 0 & -3\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}\]

Solving the first row gives us x2 =0, and that means so is x1 = 0.

And finally, we get the result of Null Space as:

\[\begin{bmatrix}x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix} \]

A Null Vector.

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