Relationship between the rows and columns of a matrix I am having trouble understanding the relatioship between rows and columns of a matrix.
Say, the following homogeneous system has a nontrivial solution.
$$
3x_1 + 5x_2 − 4x_3 = 0  \\
−3x_1 − 2x_2 + 4x_3 = 0  \\
6x_1 + x_2 − 8x_3 = 0\\
$$
Let A be the coefficient matrix and row reduce 
$\begin{bmatrix}
A & \mathbf 0
\end{bmatrix}$ 
to row-echelon form:  
$\begin{bmatrix}3&5&-4&0\\-3&-2&4&0\\6&1&-8&0\\ \end{bmatrix}
\rightarrow
\begin{bmatrix}3&5&-4&0\\0&3&0&0\\0&0&0&0\\ \end{bmatrix}$
$\quad a1 \quad a2 \quad \,a3$
Here, we see $x_3$ is a free variable and thus we can say 3rd column,$\,a_3$, is in $\text{span}(a_1, a_2)$
But what does it mean for an echelon form of a matrix to have a row of $0$'s?
Does that mean 3rd row can be generated by 1st & 2nd rows?
just like 3rd column can be generated by 1st & 2nd columns?
And this raises another question for me, why do we mostly focus on columns of a matrix?
because I get the impression that ,for vectors and other concepts, our only concern is
whether the columns span $\mathbb R^n$ or the columns are linearly independent and so on.
I thought linear algebra is all about solving a system of linear equations, 
and linear equations are rows of a matrix, thus i think it'd be logical to focus more on rows than columns. But why?

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
