Complexification Complexification $V_{\mathbb{C}} = V \oplus iV$ with elements $u + iw$ ($u, w \in V$), and $T_{\mathbb{C}}(u + iw) = Tu + iTw$. Write $p = p_T$ and $q = p_{T_{\mathbb{C...

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...

Properties Suppose $ T \in \mathcal{L}(V) $, then $ T $ is nilpotent $\iff \dim G(0,T) = \dim V $. Jordan Basis Every nilpotent operator has a Jordan basis. Let $N \in \mathcal{L}...

Translate Re-basing Suppose $ U $ is a subspace of $ V $ and $ v,w \in V $. Then \[x \in v + U \Leftrightarrow x + U = v + U.\] Corollary: Two translates of a subspace are equal or dis...

System of linear equations For $A \in \mathbf{F}^{m\times n}$ and one particular $b \in \mathbf{F}^n$, then $Ax = b$ has exactly one solution $\iff$ ($ A $ injective) and ($ b \in \operatorn...

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...

Polynomials Bézout identity Suppose $ p, q \in \mathcal{P}(\mathbb{C}) $ are nonconstant polynomials with no zeros in common. Let $ m = \deg p $ and $ n = \deg q $. There exist \(r \in \math...

Dual Space and Dual Map Suppose $V$ is finite-dimensional. Then $ V \cong V’ $, but finding an isomorphism from $V$ onto $V’$ generally requires choosing a basis of $V$. $ \dim U^0 ...

Extension and Lift: A Factoring Duality Extension (shared domain) Suppose $ W_1 $ is finite-dimensional, $ S \in \mathcal{L}(V, W_1) $ and $ T \in \mathcal{L}(V, W_2) $. Then \(\operatorname...

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. ...