# A0: Linear Algebra - Material for the year 2020-2021

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16 lectures

The core of linear algebra comprises the theory of linear equations in many variables, the theory of matrices and determinants, and the theory of vector spaces and linear maps. All these topics were introduced in the Prelims course. Here they are developed further to provide the tools for applications in geometry, modern mechanics and theoretical physics, probability and statistics, functional analysis and, of course, algebra and number theory. Our aim is to provide a thorough treatment of some classical theory that describes the behaviour of linear maps on a finite-dimensional vector space to itself, both in the purely algebraic setting and in the situation where the vector space carries a metric derived from an inner product.

Students will deepen their understanding of Linear Algebra. They will be able to define and obtain the minimal and characteristic polynomials of a linear map on a finite-dimensional vector space, and will understand and be able to prove the relationship between them; they will be able to prove and apply the Primary Decomposition Theorem, and the criterion for diagonalisability. They will have a good knowledge of inner product spaces, and be able to apply the Bessel and Cauchy--Schwarz inequalities; will be able to define and use the adjoint of a linear map on a finite-dimensional inner product space, and be able to prove and exploit the diagonalisability of a self-adjoint map.

Definition of an abstract vector space over an arbitrary field. Examples. Linear maps. [1]

Definition of a ring. Examples to include $\mathbb{Z}$, $F[x]$, $F[A]$ (where $A$ is a matrix or linear map), $\mbox{End}(V)$. Division algorithm and Bezout's Lemma in $F[x]$. Ring homomorphisms and isomorphisms. Examples. [2]

Characteristic polynomials and minimal polynomials. Coincidence of roots. [1]

Quotient vector spaces. The first isomorphism theorem for vector spaces and rank-nullity. Induced linear maps. Applications: Triangular form for matrices over $\mathbb{C}$. Cayley-Hamilton Theorem. [2]

Primary Decomposition Theorem. Diagonalizability and Triangularizability in terms of minimal polynomials. Proof of existence of Jordan canonical form over $\mathbb{C}$ (using primary decomposition and inductive proof of form for nilpotent linear maps). [3]

Dual spaces of finite-dimensional vector spaces. Dual bases. Dual of a linear map and description of matrix with respect to dual basis. Natural isomorphism between a finite-dimensional vector space and its second dual. Annihilators of subspaces, dimension formula. Isomorphism between $U^0$ and $(V/U)^\prime$. [3]

Recap on real inner product spaces. Definition of non-degenerate symmetric bilinear forms and description as isomorphism between $V$ and $V^\prime$. Hermitian forms on complex vector spaces. Review of Gram-Schmidt. Orthogonal Complements. [1]

Adjoints for linear maps of inner product spaces. Uniqueness. Concrete construction via matrices [1]

Definition of orthogonal/unitary maps. Definition of the groups $O_n, SO_n,U_n, SU_n$. Diagonalizability of self-adjoint and unitary maps. [2]

1) Richard Kaye and Robert Wilson, *Linear Algebra* (OUP, 1998) ISBN 0-19-850237-0. Chapters 2--13. [Chapters 6, 7 are not entirely relevant to our syllabus, but are interesting.]

1) Paul R. Halmos, *Finite-dimensional Vector Spaces*, (Springer Verlag, Reprint 1993 of the 1956 second edition), ISBN 3-540-90093-4. sections 1--15, 18, 32--51, 54--56, 59--67, 73, 74, 79.

[Now over 50 years old, this idiosyncratic book is somewhat dated but it is a great classic, and well worth reading.]

2) Seymour Lipschutz and Marc Lipson, *Schaum's Outline of Linear Algebra* (3rd edition, McGraw Hill, 2000), ISBN 0-07-136200-2. [Many worked examples.]

3) C. W. Curtis, *Linear Algebra - an Introductory Approach*, (4th edition, Springer, reprinted 1994).

4) D. T. Finkbeiner, *Elements of Linear Algebra* (Freeman, 1972). [Out of print, but available in many libraries.]