MA2008: Linear Algebra II

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA2008
External Subject Code 100405
Number of Credits 20
Level L5
Language of Delivery English
Module Leader Professor Alexander Balinsky
Semester Autumn Semester
Academic Year 2022/3

Outline Description of Module

This lecture-based module continues to introduce students to the fundamental notions of linear algebra.  Picking up where MA1008 Linear Algebra I left off, we interpret bases as coordinate isomorphisms with kn. We then explain how this allows us to write down and manipulate abstract linear maps as matrices. Matrix computation methods are introduced, and studied at length. The students are taught how to compute the kernel and the image of a linear map via matrices, and as an application — how to solve systems of homogeneous and non-homogeneous linear equations. Finally, we study how the matrix of a linear map changes with a change of basis and the standard form that it can be brought to.

The module concludes with some of the most exciting and challenging material in Linear Algebra.  We review the standard form that a matrix of a linear map can brought to by changing the bases of the source and the image space.  We then ask what happens in the case of an endomorphism, where only one basis can be chosen for both.  Diagonalisability is discussed, leading naturally to the notions of eigenvectors and eigenvalues.  Students are taught how to compute these via the characteristic polynomial.  The existence of non-diagonalisable linear maps is established, leading up to the main theorem of the course – the Jordan Normal Form of endomorphism.  We finish by studying dual vector spaces and dual bases, bilinear forms, orthogonality, inner products, and Gram-Schmidt orthonormalisation algorithm.

On completion of the module a student should be able to

  • Understand the basic abstract notions of linear algebra: vector space, linear map, basis, etc.
  • Understand and work with concrete examples of these abstract notions for the vector spaces of n-tuples of scalars, matrices, polynomials, etc.
  • Understand matrices as a computational tool for working with linear maps and be able to work with them in that capacity
  • Check a set of vectors for linear independence and/or spanning properties
  • Solve homogeneous and non-homogeneous systems of linear equations
  • Put the matrix of a linear map into its standard form
  • Invert a matrix
  • Compute the change-of-basis matrix and use it to transform matrices of linear maps
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How the module will be delivered

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

  • Weekly face to face classes (e.g. labs, lectures, exercise classes)
  • Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed

Skills:

 

Understand the basic abstract notions of linear algebra and be able to comnute them in concrete numerical examples

Transferable Skills:

Appreciation of abstract notions of linear algebra and their ubiquity in all branches of mathematics. Matrix manipulation: reduced row and echelon form, rank and nullity, inverting a matrix, determinant, change of basis. Solving systems of linear equations.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 100 Linear Algebra Ii 3

Syllabus content

 

  • Linear maps as matrices: a worked example. Flow chart of writing down linear maps as matrices. 

  • Bases as coordinate isomorphisms to kn. Examples. 

  • Reduction to maps kn→ km. Examples.
  • The maps kn→ km are the same as m × n matrices. Examples. 

  • Linear map composition as matrix multiplication. Examples. 

  • Row and column operations. Reduced echelon form.
  • The rank and the nullity of a linear map. The row/column 
span, rank, and nullity of a matrix. Identifying these notions.
  • The rank-nullity formula. The standard matrix of a linear map.
  • Homogeneous and non-homogeneous systems of linear equations. Their associated matrices. Linear map interpretation.
  • Solving a homogeneous system of linear equations by computing the kernel of its matrix.
  • Solving an inhomogeneous system of linear equations by the extended matrix manipulation.
  • Endomorphisms of vector spaces and their invertibility. Determinants. Invertibility of a matrix.
  • Inverting a matrix, method I: Simultaneous row operations.
  • Inverting a matrix, method II: Determinant and minors.
  • Change of basis and its associated matrix. Basis-independence of determinants.
  • Motivation: what is the standard matrix of an endomorphism?
  • Diagonalisable endomorphisms. Examples.
  • Eigenvectors, eigenvalues and eigenspaces. Diagonalisation as finding a basis of eigenvectors. Examples.
  • Computing the eigenvalues and eigenvectors. The characteristic polynomial of an endomorphism.
  • Non-diagonalisable endomorphisms. Computing generalized eigenspaces. Geometric and algebraic multiplicities.
  • The dual of a vector space. Dual bases. Examples.
  • Motivation: the usual dot product of kn.
  • Bilinear forms. Non-degenerate and symmetric bilinear forms. Interpreation as isomorphisms V W*.
  • Orthogonality. The orthogonal complement of a vector set.(U)=SpanU.
    Examples, e.g. (2 vectors)┴ = perpendicular line.
  • Inner products over R. Examples.
  • Sesqui-linear forms. Inner products over C. Examples.
  • Orthonormal bases.The Gram-Schmidt orthonormalisation. Examples.
  • Diagonalisability of normal endomorphisms on vector spaces over C. Specialisation to self-adjoint, unitary, symmetric, orthogonal matrices. Examples.
  • Primary Decomosition Theorem.The corresponding decomposition of the matrix into (almost) nilpotent diagonal blocks
  • The Jordan block decomposition of a nilpotent matrix. Jordan chains. Examples.
  • The Jordan Normal Form of an endomorphism.The algorithm for computing it. Examples.

     

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