Matrices¶
Recall (doc/postdoc). Not a first course.
Job: coordinate representations of linear maps \(T:K^n\to K^m\); composition \(=\) product.
Core¶
\(A=(a_{ij})\in K^{m\times n}\). Product \((AB)_{ik}=\sum_j a_{ij}b_{jk}\) when sizes match.
| Concept | Meaning |
|---|---|
| Linear map | \(T_A(x)=Ax\); \(T_{AB}=T_A\circ T_B\) |
| Rank | \(\dim\operatorname{Col}A=\dim\operatorname{Row}A\) |
| Kernel / image | \(\ker A=\{x:Ax=0\}\), \(\operatorname{im}A=\operatorname{Col}A\) |
| Transpose / adjoint | \((A^T)_{ij}=a_{ji}\); \(A^*\) conjugate transpose |
| Elementary ops | left mult. by elementary matrices; RREF |
Identity: \(I_n\). Block structure for products when partitions align.
Change of basis: \([T]_{B'}=P^{-1}[T]_B P\) (same endomorphism).
Used later for¶
- Determinants / inversion
- Eigenvalues: spectral decomposition
- Systems: \(Ax=b\)
- Lie groups: \(\mathrm{GL}_n\), matrix exp
Check¶
Does \(AB=I\) imply \(BA=I\) for square matrices over a field?
Yes — finite-dimensional: left inverse \(\Leftrightarrow\) right inverse \(\Leftrightarrow\) invertibility. Fails for operators on infinite-dimensional spaces.