I am having trouble understanding the relatioship between rows and columns of a matrix.
Say, the following homogeneous system has a nontrivial solution.
Let A be the coefficient matrix and row reduce [A0] to row-echelon form:
Here, we see is a free variable and thus we can say 3rd column,, is in
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 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?