Linear Algebra - Basis for a row space

[jinksys]

A =

 1  2 -1  3
 3  5  2  0
 0  1  2  1 
-1  0 -2  7

Problem: Find a basis for the row space of A consisting of vectors that are row vector of A.

My attempt:

I transpose the matrix A and put it into reduced row echelon form. It turns out that there are leading ones in every column. Therefore, the basis includes every row vector from A.

Is this the correct way to handle this problem?

 

[vela]

Yes, that works.

 

[Mark44]

What you found means that the rows of A form a basis for R⁴, meaning that every vector in R⁴ can be written as a linear combination of the four vectors that are the rows of A.