Relationship between the rows and columns of a matrix
Solution 1:
Having a row of $0$'s in the row-echelon form means that we were able to write the third row of $A$ as a linear combination of the second and first rows. As it so happens for square matrices, this is true precisely when we can write the columns as a linear combination of each other (that is, when the columns are not linearly independent). If you further reduce this to reduced row-echelon form, you get $$\begin{bmatrix} 1 & 0 & -4/3 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Because the third column lacks a pivot, $x_3$ is our free variable, which means that we can write $a_3$ as a linear combination of the other two columns.
There's a very good reason for focusing on the columns of a matrix. This comes out of using $A$ as a linear transformation, where the "column space" gives us the "range" of the function $f(\vec x) = A \vec x$.
Solution 2:
Excellent question.
In some sense, the equations and variables represent equivalent information. Sometimes it is easier to approach the problem from the point of view of rows-equations-constraints, and sometimes - from the point of view of variables.
This is very deeply related to the notion of the dual problem in linear programming. That is exactly what converts equations/constraints to rows and vice versa, looking at a different problem.
Mathematically, this boils down to either working with the matrix $A$ or with some form of $A^T$, which converts rows into columns and columns into rows. Not surprisingly, the main characteristic properties for $A$ and $A^T$ are the same, like rank, eigenvalues/singular values, determinant, trace, etc.
Two reference links on duality:
- Wikipedia article on duality
- MIT notes on duality in Linear Programming