From this answer, I see that, given a list of vectors, we can put them as the columns of a matrix, then reduce it, and the pivots will tell us which of the original vectors form a basis for the span of those vectors.
However, I'm wondering, can't we just put the list of vectors as the rows of a matrix, then reduce it, and find a basis this way? Since row operations don't change the row space, this method should also work. We just look at the rows that have a leading variable.
Though, I can see that the first method of putting the vectors in a column might be useful if we want to find out which of the original vectors form a basis.
Best Answer
[I’m sure that I’ve answered this before, but can’t find anything relevant at the moment.]
Yes, both methods will get you a basis for the span of the vectors, and your observation is correct: writing them as columns of a matrix will let you find a linearly-independent subset of the original vectors that has the same span. On the other hand, writing them as rows generally gives you a “nicer” basis: The first part of each vector in the basis will consists of a bunch of zeros with a single one somewhere. Which one is preferable depends entirely on what you’re then going to do with that basis.