# Find the basis for the row space

• mottov2
In summary, the row space is the span of the rows of a matrix and can be found by using row reduction techniques to transform the matrix into reduced row echelon form. Finding the basis for the row space allows us to identify the linearly independent rows, which can be helpful in solving systems of linear equations and understanding the properties of the matrix. The dimension of the row space is equal to the number of linearly independent rows and can provide important information about the rank of the matrix and the number of solutions to a system of linear equations. It is possible for different matrices to have the same row space, but the actual rows in the row space may be different.
mottov2

## Homework Statement

Find the basis for the row space

## The Attempt at a Solution

the given matrix is

0 1 2 1
2 1 0 2
0 2 1 1

So i reduced to row-echeleon form

2 1 0 2
0 1 2 1
0 0 3 1

so then rank = 3. My textbook states that the basis of the row space are the row vectors of leading ones, hence the basis for row space is
{ [2,1,0,2], [0,1,2,1], [0,0,3,1] }

however my textbook has this answer
{ [6,0,0,5], [0,3,0,1]. [0,0,3,1] }

HOW?!??!??! :-(

edit: NVM figured it out. Didnt realize the textbook reduced the matrix to reduced row form.

Last edited:
also worth noting any linearly independent set of vectors that spans the space is a basis. In general there is no unique basis

## What is the row space?

The row space is the span of the rows of a matrix, meaning it is the set of all possible linear combinations of the rows of the matrix.

## Why is it important to find the basis for the row space?

Finding the basis for the row space allows us to identify the linearly independent rows of a matrix, which can be useful in solving systems of linear equations and understanding the properties of the matrix.

## How do I find the basis for the row space?

To find the basis for the row space, you can use row reduction techniques to transform the matrix into reduced row echelon form. The non-zero rows in the reduced matrix form the basis for the row space.

## What is the significance of the dimension of the row space?

The dimension of the row space is equal to the number of linearly independent rows in the matrix. This can tell us important information about the rank of the matrix and the number of solutions to a system of linear equations.

## Can the row space be the same for different matrices?

Yes, it is possible for two different matrices to have the same row space if they have the same number of linearly independent rows. However, the actual rows in the row space may be different.

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