Here we have a matrix X which consists of the columns of A. So, in the question given: the rows of AT (Transpose of A) are the columns of A.
So, we have the basis of C which consists of the first two transposed rows of RRE(A).
Now, from what I know: N(A) i.e. the null space of A directly gives the linear combinations of the columns of A (i.e. shows any linear dependency between them). Now, we need to find the linear combinations of columns of A using the rows of At. N(At) i.e. the null space of A gives the linear combinations of the columns of At (i.e. rows of A – not columns of A). So, this won't help finding the link for the columns of A?
So, I need to find the rows of At or the columns of A as a linear combination of the first two rows of RRE(At). I can do this by inspection, but I want to know .... in general: How can you find the linear dependence of rows of a matrix using its RRE(A) i.e. reduced row echelon form?
Like, you can find the linear dependence of columns of A using N(A) : Ax = 0, how do you do that for RRE(A)?
Is there a link, and what is the use of RRE(A) for checking for linear dependence (if possible) e.g. c3, c4, ..., cn in terms of the basis vectors e.g. c1, c2?