本科高年级Algebra计划中

Advanced linear algebra

Inner product spaces and the spectral theorem, dual spaces, Jordan and rational canonical forms, multilinear algebra and tensor products of vector spaces.

math.algebra.advanced-linear-algebra

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前置知识

Linear algebra Algebraic structures: monoids, groups and rings

22 项必需的前置知识（一直追溯到底）

你将学会

Prove the spectral theorem for real symmetric and complex normal operators.
Compute Jordan normal forms and minimal polynomials.
Define dual spaces and tensor products by universal properties and compute with bases.

定义

An inner product on a real vector space is a symmetric, positive-definite bilinear form.
The dual space V* is the space of linear maps from V to the field k.
The tensor product V ⊗ W comes with a bilinear map V × W → V ⊗ W through which every bilinear map out of V × W factors uniquely via a linear map.

定理

Spectral theorem: every real symmetric matrix is orthogonally diagonalisable.
Over an algebraically closed field every square matrix is similar to a Jordan normal form, unique up to the order of the blocks.
Cayley–Hamilton: a square matrix satisfies its own characteristic polynomial.
dim(V ⊗ W) = dim V · dim W, and for finite-dimensional V the map V → V** is a natural isomorphism.

常见误解

Thinking V ≅ V* canonically; such an isomorphism depends on a choice (of basis or inner product), whereas V ≅ V** is natural.
Believing every element of V ⊗ W has the form v ⊗ w.

通向

Modules Representation theory of finite groups Smooth manifolds Functional analysis

可选发现

Natural transformations The isomorphism V → V** is natural in V while V ≅ V* is not; making this distinction precise was a motivation for Eilenberg and Mac Lane's definition of natural transformation.
Monoidal categories Vector spaces with the tensor product form a symmetric monoidal category.
← The axiom of choice and its equivalents

学习路径

Graduate preparation