Formula Bank
Every section page keeps its formulas in a small file, and this page collects them. Each rule says when it holds, because the conditions are where the edge cases live.
Linear Algebra
Matrix Multiplication AB
| Rule | Statement | Holds when |
|---|---|---|
| Size | \((m \times n)(n \times p)\) gives \(m \times p\) | The inner sizes agree |
| Entry | \((AB)_{ij} = \sum_k a_{ik} b_{kj}\) | Always |
| Columns | Column \(j\) of \(AB\) is \(A\) times column \(j\) of \(B\) | Always |
| Rows | Row \(i\) of \(AB\) is row \(i\) of \(A\) times \(B\) | Always |
| Associative | \((AB)C = A(BC)\) | Always |
| Transpose | \((AB)^{\mathsf T} = B^{\mathsf T} A^{\mathsf T}\) | Always |
| Order | \(AB = BA\) | Only for special pairs, not in general |
| Zero product | \(AB = 0\) forces \(A = 0\) or \(B = 0\) | Never in general |