u/Hawexp

I'm learning about pivots and free variables in linear algebra, and it seems like while the number of pivots is inherent to the system, the choice of which variables are labeled the pivots isn't?

For example, if we have:

x + y = z

Then we might be able to say that x and y are free variables and together determine z, but I could just as well see that if we choose y and z, x is determined as well.

Is it generally true that any combination of n variables where n is equal to the number of pivots, works as choices for the pivots?

I'm a young adult who's new to investing. I have about 15k dollars saved up from an internship I did a few years ago. It's just sitting in a savings account right now. From what I've read, it'd be best to put it in the S&P 500 since I don't have a use for it anytime soon - I'm fine with letting it sit for decades if I know it'll outdo my savings account growth.

Is this advice still sound? I admit I'm not too familiar with why this advice is considered good, so I have a poor framework for analyzing it. All I really know is that the S&P 500 is a list of the 500 most valuable companies, or something to that effect, and that it's considered a near certainty that its value outpaces inflation asymptotically.