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Linear Algebra II:  2008-2009

Term

Hilary Term 2009  (8 lectures)

Learning outcomes

Students will

  1. have an understanding of matrices, and their representation of linear transformations of vector spaces, including change of basis.
  2. understand the elementary properties of determinants.
  3. understand the elementary parts of eigenvalue theory with some of its applications.

Synopsis

Review of a matrix of a linear transformation with respect to bases, and change of bases. [1 lecture]

Permutations of a finite set, composition of permutations. Cycles and standard cycle-product notation. The result that a permutation is a product of transpositions. The parity of a permutation; goodness of the definition; how to calculate parity from cycle structure.

Determinants of square matrices; properties of the determinant function; determinants and the scalar triple product.[2 lectures]

Computation of determinant by reduction to row echelon form.

Proof that a square matrix is invertible if and only if its determinant is non-zero.

Determinant of a linear transformation of a vector space to itself. [2$ \frac{1}{2} $ lectures]

Eigenvalues of linear transformations of a vector space to itself. The characteristic polynomial of a square matrix; the characteristic polynomial of a linear transformation of a vector space to itself. The linear independence of a set of eigenvectors associated with distinct eigenvalues; diagonalisability of matrices. [2$ \frac{1}{2} $ lectures]

Reading list

  1. C. W. Curtis, Linear Algebra - An Introductory Approach (Springer, 4th edition, reprinted 1994).
  2. R. B. J. T.  Allenby, Linear Algebra (Arnold, 1995).
  3. T. S.  Blyth and E. F.  Robertson, Basic Linear Algebra (Springer, 1998).
  4. D. A.  Towers, A Guide to Linear Algebra (Macmillan, 1988).
  5. D. T.  Finkbeiner, Elements of Linear Algebra (Freeman, 1972). [Out of print, but available in many libraries]
  6. B. Seymour Lipschutz, Marc Lipson, Linear Algebra (McGraw Hill, Third Edition 2001).
    For permutations:
  7. R. B. J. T. Allenby, Rings, Fields and Groups (Edward Arnold, Second Edition,1999). [Out of print, but available in many libraries also via Amazon]
  8. W.B.S. Stewart, Abstract Algebra (Mathematical Institute Lecture Notes, 1994).

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