ONE way to interpret the linear problem A~x = ~b for ~x is that we wish to write ~b as a

linear combination of the columns of A with weights given in ~x. This perspective does not change when we allow A ∈ Rm×n to be non-square, but the solution may not exist or be unique depending on the structure of the column space of A. For these reasons, some techniques for factoring matrices and analyzing linear systems seek simpler representations of the column space of A to address questions regarding solvability and span more explicitly than row-based factorizations like LU.