Advanced linear algebra

Builds on

Required prerequisites, all the way down: 22

You will learn to

• 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.

Definition

• 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.

Theorem

• 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.

Common misconceptions

• 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.

Leads to

Optional discoveries

• 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

Learning pathways

• Graduate preparation

Further reading

• Sheldon Axler, Linear Algebra Done Right (1996)
• Paul R. Halmos, Finite-Dimensional Vector Spaces