Systems of Equations¶
Recall (doc/postdoc). Not a first course.
Job: \(Ax=b\) for \(A\in K^{m\times n}\); solution set is empty, a point, or affine subspace of \(\ker A\).
Core¶
\[
\begin{cases}a_{11}x_1+\cdots+a_{1n}x_n=b_1\\ \vdots\\ a_{m1}x_1+\cdots+a_{mn}x_n=b_m\end{cases}
\iff Ax=b
\]
| Method | Algebraic content |
|---|---|
| Substitution | eliminate variables |
| Elimination | elementary row ops \(\sim\) left mult. by \(E\in\mathrm{GL}\) |
| Matrix | RREF; consistency via augmented rank |
Structure theorem (field \(K\)): solutions exist iff \(\operatorname{rank}A=\operatorname{rank}[A|b]\); then affine space of dim \(n-\operatorname{rank}A\).
Homogeneous \(Ax=0\): always \(0\in\) solution; basis of \(\ker A\) = nullspace basis.
Used later for¶
- Matrices / rank–nullity
- Optimization: linear constraints
- Differential equations: linear const-coeff systems
- Algebraic geometry: complete intersections / regular sequences (nonlinear)
Check¶
If \(A\) is \(n\times n\) over a field and \(\det A=0\), can \(Ax=b\) still have solutions?
Yes — when \(b\in\operatorname{Col}A\); infinitely many if so ( nontrivial ker). Empty if \(b\notin\operatorname{Col}A\).