Notation Symbol Meaning $V$ a vector space over $\mathbf{F}$ (nonzero, finite-dimensional unless stated) $E(\lambda,T) = \operatorn...

This post continues Nilpotent Operators. It has two halves. The first collects what can be said about commuting operators in general, and ends with a reduction: for a complex vector space, the whol...

Notation Symbol Meaning $\mathcal{L}(V)$ linear operators $V \to V$ $\mathcal{M}(T)$ matrix of $T$ with respect to a chosen b...

Notes on eigenvalues, eigenspaces, generalized eigenspaces, and the ladder of normal forms that sits on top of them: upper-triangular $\to$ primary decomposition $\to$ Jordan $\to$ diagonal. Notat...

Notation Symbol Meaning $\mathbf{F}$ the field $\mathbb{R}$ or $\mathbb{C}$ $V$ a vector space over $\mathbf{F}$ (nonzero, fi...

The transpose identities used below — $\operatorname{null} T’ = (\operatorname{range} T)^0$ and $\operatorname{range} T’ = (\operatorname{null} T)^0$ — are proved in Dual Space. Notation ...

Notation Symbol Meaning $V$ a vector space over $\mathbf{F}$ (nonzero, finite-dimensional unless stated) $\mathcal{L}(V,W)$ l...

Notation Symbol Meaning $V$ a vector space over $\mathbf{F}$ (nonzero, finite-dimensional unless stated) $\mathcal{L}(V)$ lin...

Notation Symbol Meaning $\mathbf{F}$ the field $\mathbb{R}$ or $\mathbb{C}$ $\mathcal{L}(V,W)$ linear maps $V \to W$ ...

Affine subspace affine = linear structure with the origin forgotten An operation on points commutes with every translation $ x \mapsto x + t $ if and only if its coefficients sum to 1. ...