Cfrgtkky

I am trying to understand how to check whether the vectors belong to a span or not. However, I am confused with the difference in solutions between the determination of the span of the vectors made of 2x1 and 1x2 matrices. In the solutions to my textbook problems, I encounter with two different solutions.
For 1x2 matrices, like [1 2] and [-1 1], it is written that

c_1 v_1 + c_2 v_2 = v

I am okay with that, but in the next step the vectors which are 1x2 matrices are converted to 2x1 matrices and the solution goes on like (in augmented matrix)

1 -1 . a

2 1 . b

and it is being checked whether it is consistent or not. I do not get why we are converting our 1x2 matrices to 2x1 matrices.

After that, I checked for 2x1 matrices. This time, the solutions starting with defining a 2x1 vector v with all the terms equal to 1. Then, it continues with writing the 2x1 vector and also the new v vector in an augmented matrix where v is in the augmented part. Then, it ends up similarly with checking the matrix is consistency.

I know that giving 1 to every term is an arbitrary thing, but I did not get the logic behind them all.

Thanks