# Why are spans in Linear Algebra represented using free variables?

I have been doing some refresher work in Khan Academy for Linear Algebra and there was mention of how a column space or any other space can be represented as a fucntion of the free variables and not the other way around.

I am curious about why that would be the case? And would it be the only solution?

I can see it might be a little easier to do it this way since linear models come in a format better suited to such a representation. But would it be wrong to represent it the other way?

Is there really an inherent correctness in such a representation?

• I don't understand what you mean by "and not the other way around" Commented Sep 7, 2020 at 14:43
• In this video from KA we find the pivot vectors of a matrix and its associated variables/constants X are put in an equation. X1 is associated with the pivot vector, except X1 is taken as a function of the other variables. Not the other way around. Which is something that I am still confused by Commented Sep 7, 2020 at 14:46
• Ok. In other words, the pivot variable is taken as a function of the other (free) variables and not the other way around. Commented Sep 7, 2020 at 14:50
• You might find my post here to be helpful Commented Sep 7, 2020 at 14:51
• I took a second to read through your answer and I think it makes sense to me now. Since the pivots are expected to be independent(if I can use a colloqial term, foundational) with defining a space, then we want to have a basis be built around our pivot variables so that all the other points in space are just a reconstitution of these pivot variables. So we we do pivot variable=equation of other variables... What we are actually doing is representing our spanning vectors as just the equation that represents the pivot vecors Commented Sep 7, 2020 at 14:58