Matrices and pivoting

Pivoting is using non zero numbers to manipulate the form of a matrix, for example when putting it into echelon form (EF) and eventually reduced echelon form (REF) if needed. Yay. Wasn't that (not) helpful?

(some books will say row echelon form and reduced row echelon form--don't sweat it)

Good news, there is more to it. The pivots of the matrix (those leading ones in REF) tell us where our pivot columns are (you can actually find them in EF, but REF gives us some more info that may be useful. Knowing our pivot columns tells us what column vectors in our column space are linearly independent and which ones are linearly dependent. It tells us the rank (dimensionality) of our column space. It tells us what the basis (unit) vectors of that column space are (how the space is skewed, rotated, or transformed and what vectors we can use to reach any point of it).

Pivoting also plays a role in linear optimization (also called linear programming) with similar ideas. There may be more areas, but I haven't learned about them.

Real world use is any problem you can describe with matrices. Problems in physics, operations research, engineering, logistics (supply chain), computer graphics, computational spatial systems, etc.

Another way of figuring out real world uses is to consider the types things they can represent. If you have learned about pivoting you have learned about systems of equations. They can be used to represent positions (a photo can be represented as a matrix, so can a character of text), they can be used to represent connections between nodes in networks. They can be used to represent a space or a changes to any of the things so far.

Finding pivots has been automated. You can write your own code for fun, but there are people that specialize in creating highly efficient and accurate algorithms for computing such things taking hardware and software constraints into consideration. If you were to work with matrices directly and professionally you will likely have a large selection of software tools to choose from.

/r/math Thread