MATH 2574H Lecture Notes - If And Only If
Document Summary
Theorem: a linear system is consistent if and only if (iff) an echelon form of the associated augmentedg matrix has no row of the form [0000b] (b nonzero) If the system is consistent, unique sol"n (if no free variables) or infinitely many (if there = at least 1 free variable) Def: a homogeneous linear system is one of the form. Remark: all homogeneous systems are consistent (bi are all 0) and x1 = x2 = xn = 0 is always a solution (where all of them = 0 is a trivial solution) -> either infinite solutions or just trivial soluitons. Theorem: suppose a is an n by n matrix, then the homogenous system w/ coefficient matrix a has only the trivial solution iff a is row equivalent ot the identity matrix) Ex: x1 + 2 by solving it, you get identity matrix --> only trivial solution.
