## Introduction to Column Space Calculator:

Column Space Calculator is an online tool that helps you **calculate column space** of a given matrix in a run of time. It evaluates the square matrix and provides you with solutions in which all the possible linear combinations of a matrix are involved.

The col space calculator is a beneficial tool for those who want to evaluate the column space of a matrix problem without doing manual calculations and gives you a solution.

## What is the Column Space of Matrix?

Column Space of a matrix is a process that is used to find all the **linear combinations** of a matrix that span the entire system in linear algebra.

For a column space of a matrix A (in which vectors are a1,a2,a3,..., an) that are multiplied with a scalar value (x1,x2,x3,..xn) to span the subdomain R. It is denoted by the col(A). The pivot point is the basis that is linearly independent of that matrix A. The formula used by the column space calculator is,

$$ col(A) \;=\; \{ x_1 \overrightarrow{a_2} + x_2 \overrightarrow{a_2} + … + a_n \overrightarrow{a_n} | x_1, x_2, … , x_n ∈ \mathbb{R} $$

## Working Process of Col Space Calculator:

The column space of a matrix calculator uses the easiest method to solve the column space of a given matrix immediately. Due to its advanced server, you can give complicated matrix as an input in it and it gives you a solution. Let's see the **evaluation steps** of the column space matrix problem.

**Step 1**:

Identify the given matrix A.

**Step 2**:

To make the row echelon form (REF) or reduced row echelon form (RREF) from the given matrix apply the row reduction (Gaussian elimination) method.

**Step 3**:

After achieving the reduced echelon form identify the pivot columns in the diagonal elements in a matrix

**Step 4**:

The columns in the original matrix A corresponding to the pivot columns form the basis of the column space. These basis spans are column space.

## Calculate Column Space: An Example

Let's see an **example** of a column space matrix solution to understand how the column space calculator calculate column space in steps.

### Example: Find Colume Space

$$ \left[ \begin{matrix} 1 & 2 & 3 & 2 \\ 3 & 0 & 1 & 8 \\ 2 & -2 & -2 & 6 \\ \end{matrix} \right] $$

**Solution**:

$$ \left[ \begin{matrix} 1 & 2 & 3 & 2 \\ 3 & 0 & 1 & 8 \\ 2 & -2 & -2 & 6 \\ \end{matrix} \right] $$

Use the Gauss elimination method to achieve row echelon form,

$$ Interchanging\; rows\; R_1 \leftrightarrow R_2 $$

$$ =\; \left[ \begin{matrix} 3 & 0 & 1 & 8 \\ 1 & 2 & 3 & 2 \\ 2 & -2 & -2 & 6 \\ \end{matrix} \right] $$

$$ R_2 \leftarrow R_2 - 0.3333333333 \times R_1 $$

$$ =\; \left[ \begin{matrix} 3 & 0 & 1 & 8 \\ 0 & 2 & 2.6666666667 & -0.6666666667 \\ 2 & -2 & -2 & 6 \\ \end{matrix} \right] $$

$$ R_3 \leftarrow - 0.6666666667 \times R_1 $$

$$ =\; \left[ \begin{matrix} 3 & 0 & 1 & 8 \\ 0 & 2 & 2.6666666667 & -0.6666666667 \\ 0 & -2 & -2.6666666667 & 0.6666666667 \\ \end{matrix} \right] $$

$$ R_3 \leftarrow R_3 + R_2 $$

$$ =\; \left[ \begin{matrix} 3 & 0 & 1 & 8 \\ 0 & 2 & 2.6666666667 & -0.6666666667 \\ 0 & 0 & 0 & 0 \\ \end{matrix} \right] $$

The rank of the matrix of all non-zero rows of a matrix is 2.

**For Column Space**:

The matrix has 2 pivots and pivot values in the columns are 3 and 2.

The Column Space is that span the given matrix.

$$ \left[ \begin{matrix} 1 \\ 3 \\ 2 \\ \end{matrix} \right],\; \left[ \begin{matrix} 2 \\ 0 \\ 2 \\ \end{matrix} \right] $$

## How to Use Column Space Calculator?

The col space calculator has a user-friendly layout that allows you to solve various types of matrix column space problems. You just need to put your problem in this calculator and follow some guidelines so you can get results without any inconvenience. These guidelines are:

**Choose the size**of the matrix for the column space matrix.- Enter the elements of your matrix in the input field for the column space solution.
- Recheck your given input matrix value before clicking on the calculate button of the column space basis calculator to get the solution of the column space question.
- Click on the “Calculate” button for the solution column space problems.
- If you want to check the working process of the column space matrix calculator then use the load example for the calculation to get an idea about its solution.
- The “Recalculate” button allows you to evaluate more examples of column space problem solution

## Output of Column Space of a Matrix Calculator:

Column Space Calculator provides you with a column space problem solution (both real or complex numbers) as per your input 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 column space** problem

**Steps Box**

Click on the steps option so that you get the solution of column space questions in a step-by-step method

## Why you Choose our Column Space Basis Calculator?

The column space of matrix calculator has different features whenever you use it to solve column space problems to get the solution. Our tool only gets the input value of a matrix and gives a column space solution without any restriction. These features will compel users to use it for evaluation which are:

- The column space matrix calculator is a trustworthy tool as it always provides you with
**accurate solutions**of column space problem. - It is a swift tool that evaluates column space problems with solutions in a couple of seconds.
- Column space of a matrix calculator is a learning tool that helps children about the concept of column space of a matrix very easily on online platforms without going to any tutor.
- It is a handy tool that solves column space matrix problems quickly because you do not put any type of external effort for calculation.
- Col space calculator is a free tool that allows you to use it for the calculation of column space matrix without getting any fee.
- It is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of column space matrix problems.
- Column space calculator can operate on a desktop, mobile, or laptop through the internet to solve column space problems.