This Row Space Calculator is tool for linear algebra students and practitioners that calculates the row space of a matrix. This calculator shows the step-by-step calculation of the row space of matrix.

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To use this Row Space Calculator, follow the steps below:

- Enter the number of rows in first input box
- Enter the number of columns in second input box
- Enter the matrix elements in the input boxes provided
- Click on the "Calculate" button to get the row space of the matrix

If you find any error in the calculation, please send use the feedback form below the calculator.

The row space of a matrix is the set of all possible combinations of its rows. It's like taking different amounts of each row and adding them together.

For example, if you have a matrix M

```
\(\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{bmatrix}
\)
```

If you reduce the above matrix using rref calculator, you will get

```
\(\begin{bmatrix}
1 & 0 & -1 \\
0 & 1 & 2 \\
0 & 0 & 0 \\
\end{bmatrix}
\)
```

Now the row space for a matrix M is the set of all possible linear combination of it rows. In this case, the row space of matrix M consists of all the vectors that can be formed as linear combinations of the rows in its reduced form.

The non-zero rows of the reduced matrix above is

```
\(\begin{bmatrix}
1 & 0 & -1 \\
0 & 1 & 2 \\
\end{bmatrix}
\)
```

These two rows are linearly independent and span the row space of the matrix M.

So the row basis of the matrix M is

```
\(\{ \begin{bmatrix} 1 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 0 & 1 & 2 \\ \end{bmatrix} \}
\)
```

To find the row space of a matrix, we can use the method called row reduction. This is the method for simplifying the matrix to its reduced row echelon form (rref).

There are various operation to reduce the rows to its simplest form like swapping the rows, multiplying the rows by a scalar and adding one row to another.

When a matrix is simplfied to its reduced form, it becomes easier to perform various operations on it. Matrix data like its rank, determinants, eigenvalues, etc can be easily calculated in its reduced form.

The row space of matrix is used for simplifying matrix to use in different components of engineering, physics and computer science.

To find the row basis of the matrix, reduce the matrix to rref (reduced row echelon form) and take the non-zero rows of the reduced matrix. These non-zero rows are the basis of the row space of the matrix.

The row space of a matrix gives information about the possible linear combinations of its rows. It is used to find the rank of the matrix, which is the number of linearly independent rows in the matrix.

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