Jordan Basis Every nilpotent operator has a Jordan basis. Let $N \in \mathcal{L}(V)$ be nilpotent. Fix a Jordan basis: there are vectors $v_1,\dots,v_k$ (the tops) and lengths $m_1,\dots,m...

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...

Invariant Subspaces Suppose $ T \in \mathcal{L}(V) $ and $ U $ is a subspace of $ V $ invariant under $ T $. Then $ U $ is invariant under $ p(T) $ for every polynomial $ p \in \mathcal{P}(\mat...

Polynomials $ T \in \mathcal{L}(V) $ $ \ q \in \mathcal{P}(\mathbf{F}) $ Minimal polynomial $ T/U $ $ q(T/U) = q(T)/U $ $ p_{T/U} \mid ...

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. ...

3E 11. Suppose $ U = \{ (x1,x2,\dots) \in \mathbf{F}^\infty : x_k \ne 0 \ \text{for only finitely many k} \} $. (b) Prove that $ \mathbf{F}^\infty/U $ is infinite-dimensional. Goal and str...

Vector Space Definition Motivation: properties of addition and scalar multiplication in $ \mathbf{F}^n $ flowchart LR subgraph addGroup["$$u + v$$"] direction TB addSet["abelian group under...