本科高年级Algebra课程已上线
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
前置知识
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
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
延伸阅读
- Sheldon Axler, Linear Algebra Done Right (1996)
- Paul R. Halmos, Finite-Dimensional Vector Spaces