Vectors and Matrices

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VECTORS AND MATRICES

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Complex numbers

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• Review of complex numbers, including complex conjugate, inverse, modulus, argument and Argand diagram. Informal treatment of complex logarithm, n-th roots and complex powers. de Moivre’s theorem. [2]

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 Vectors

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• Review of elementary algebra of vectors in R3, including scalar product. Brief discussion of vectors in Rn and Cn; scalar product and the Cauchy–Schwarz inequality. Concepts of linear span, linear independence, subspaces, basis and dimension.
• Suffix notation: including summation convention, δij and ijk. Vector product and triple product: definition and geometrical interpretation. Solution of linear vector equations. Applications of vectors to geometry, including equations of lines, planes and spheres. [5]

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Matrices

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• Elementary algebra of 3×3 matrices, including determinants. Extension to n×n complex matrices. Trace, determinant, non-singular matrices and inverses. Matrices as linear transformations; examples of geometrical actions including rotations, reflections, dilations, shears; kernel and image, rank–nullity theorem (statement only). [4]
• Simultaneous linear equations: matrix formulation; existence and uniqueness of solutions, geometric interpretation; Gaussian elimination. [3]
• Symmetric, anti-symmetric, orthogonal, hermitian and unitary matrices. Decomposition of a general matrix into isotropic, symmetric trace-free and antisymmetric parts. [1]

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Eigenvalues and Eigenvectors

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• Eigenvalues and eigenvectors; geometric significance. [2]
• Proof that eigenvalues of hermitian matrix are real, and that distinct eigenvalues give an orthogonal basis of eigenvectors. The effect of a general change of basis (similarity transformations). Diagonalization of general matrices: sufficient conditions; examples of matrices that cannot be diagonalized. Canonical forms for 2×2 matrices. [5]
• Discussion of quadratic forms, including change of basis. Classification of conics, cartesian and polar forms. [1]
• Rotation matrices and Lorentz transformations as transformation groups. [1]

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Appropriate books

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• Alan F Beardon Algebra and Geometry. CUP 2005(pdf)
• Gilbert Strang Linear Algebra and Its Applications. Thomson Brooks/Cole, 2006(pdf)
• Richard Kaye and Robert Wilson Linear Algebra. Oxford science publications, 1998(pdf)
• D.E. Bourne and P.C. Kendall Vector Analysis and Cartesian Tensors. Nelson Thornes 1992(pdf)
• E. Sernesi Linear Algebra: A Geometric Approach. CRC Press 1993(pdf)
• James J. Callahan The Geometry of Spacetime: An Introduction to Special and General Relativity. Springer 2000(pdf)

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Other Books

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• Linear Algebra with Applications( PDF,Partial Student Solution Manual,)

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Informal description of Vectors and Matrices

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The course starts with revision of complex numbers. It then introduces some more advanced ideas,including de Moivre’s theorem which may be new to you. It moves on to generalise to higher (possible complex) dimensions the familiar idea of a vector. A very important tool, suffix notation, is used for vector algebra. This is followed by the application of vector methods to geometry.

The remainder of the course is taken up with matrices: algebraic manipulation; applications to solution of simultaneous equations; geometrical applications; and eigenvectors and eigenvalues.

The material in this course is absolutely fundamental to nearly all areas of mathematics.

Learning outcomes

By the end of this course, you should:

•  be able to manipulate complex numbers and be able to solve geometrical problems using complex numbers;
• be able to manipulate vectors in R^3
  (using suffix notation and summation convention where appropriate), and to solve geometrical problems using vectors;
• be able to manipulate matrices and determinants, and understand their relation to linear maps and systems of linear equations;
• be able to calculate eigenvectors and eigenvalues and understand their relation with diagonalisation of matrices, and canonical form.