Linear Equations¶
Recall (doc/postdoc). Not a first course.
Job: solve \(ax+b=0\) in a field; prototype for \(1\times1\) linear systems and affine hyperplanes.
Core¶
Over field \(K\): \(ax+b=c\) \(\Leftrightarrow\) \(ax=c-b\).
| Case | Solution set |
|---|---|
| \(a\neq0\) | unique \(x=a^{-1}(c-b)\) |
| \(a=0\), \(b=c\) | all of \(K\) (identity) |
| \(a=0\), \(b\neq c\) | \(\emptyset\) |
Affine form: \(\{x:ax=d\}\) is a coset of \(\ker(a\cdot)\) in \((K,+)\).
Clearing denominators: work in \(K\); if the equation came from a rational expression, exclude poles.
Over rings (e.g. \(\mathbb{Z}/n\mathbb{Z}\)), uniqueness fails when \(a\) is not a unit — use annihilators / Smith form later.
Used later for¶
- Systems: \(Ax=b\)
- Matrices: row reduction, rank
- Inequalities: half-spaces \(ax\le b\)
- Algebraic geometry: hyperplanes \(V(\ell)\)
Check¶
Over \(\mathbb{Z}/6\mathbb{Z}\), how many solutions can \(2x=4\) have?
\(x\equiv2\) and \(x\equiv5\pmod6\) — two solutions; uniqueness fails when \(a\notin(\mathbb{Z}/6\mathbb{Z})^\times\).