Notes on inner products: the definition over $\mathbb{R}$ and $\mathbb{C}$, the parallelogram equality and which norms come from an inner product, and two applications of the Cauchy–Schwarz inequal...

Two operators are similar when they are the same operator seen in different coordinates. This post makes that precise through change of basis, then asks the natural question: given $S$ and $T$, how...

This post continues Nilpotent Operators and Eigenvalues and Eigenvectors, and asks which operators commute with a given $T \in \mathcal{L}(V)$. The first part collects what can be said about commut...

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

Notes on eigenvalues, eigenspaces, and generalized eigenspaces, organized around one theme: how the factorization of the minimal polynomial determines the structure of an operator. Along the way: i...

Notation Symbol Meaning $n = \dim V$ dimension of $V$ $\mathcal{L}(V)$ linear operators $V \to V$ $\mathcal{M...

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...