Taking things for granted doesn't have any fun in mathematics.
When we're computing the row space of a matrix, we're using row reduction, which doesn't really change the rowspace of a matrix, but it doesn't change the column space, and again, row reductions don't change the linear independence of the columns of a matrix, but they do change the linear independence of the rows of a matrix. AND! The basis of the row space of the matrix can be found by performing row operations on the transpose of a matrix. Now, these are the basis of the row space but the operations are done on a completely different set of entries and still we find the basis of the row space because, most probably(with mere analogy, as there is probably no theorem for this), this time around the linear dependence of the rows of the matrix is not affected.
Summarizing my large ocean of confusions to four major issues:
- First of all the problem arises with books having no reasons to whatever they explain, and it starts with not defining why we do row operations at all when we actually change the columspace of a matrix?
Then, it continues down with, why do we do row operations? And not column operations? Why isn't there a Reduced Column Echelon Form? As I've explained above, for finding the basis of the row space, it's almost that we're doing Reduced Column Echelon Form, but our doing so doesn't change the results, but there's no theorem for stating that's valid(Atleast, uptill where I've studied), I mean it doesn't change the invertability of a matrix, but again, not all matrices are invertible.
And there's more, Why does the column space of a matrix change while we're doing row operations but the linear independence doesn't change but exactly opposite with the row space, i.e. the row space doesn't change but the linear dependence change. Come to think of it, at this last point, if the linear dependence has changed, then the basis that span the row space has changed, but How come the row space doesn't change?
- What's the connection of row space with column space, and what's the geometrical interpretation of a row space?