# Elementary Linear Algebra

Recently there have been many notes about problems involving linear
independence, spanning, and row-reducing a matrix. Perhaps some
useful.

Suppose that we have matrices where: A is nXn, B is nXp, C is pX1, and
A is invertible. For example, A might be the product of elementary
matrics that you could accumulate while row reducing B, i.e. U=(A*B)
is row-reduced. And C might represent a linear dependence relation
among the columns of B, i.e. B*C=0.
By the associative property, (A*B)*C = A*(B*C) for any compatible
A,B,C.
So, when A is invertible, C represents a linear relation on the
columns of B iff C also represents a linear relation on the columns of
U=(A*B).
So, for example, columns 1,2,4,7 of B are linearly indep. iff columns
1,2,4,7 of U are lin. indep., which would be obvious if 1,2,4,7
happened to be pivot columns.