M1: Linear Algebra II (2023-24)
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- Lecturer: Profile: Richard Earl
Course information
Course Term: Hilary
Course Lecture Information: 8 lectures
Course Overview:
Linear algebra pervades and is fundamental to algebra, geometry, analysis, applied mathematics, statistics, and indeed most of mathematics. This course builds on Linear Algebra I, with a focus on how linear transformations can be understood from different geometric, algebraic and spectral perspectives.
Learning Outcomes:
Students will:
(i) understand the elementary theory of determinants;
(ii) understand the beginnings of the theory of eigenvectors and eigenvalues and appreciate the applications of diagonalizability.
(iii) understand the Spectral Theory for real symmetric matrices, and appreciate the geometric importance of an orthogonal change of variable.
(i) understand the elementary theory of determinants;
(ii) understand the beginnings of the theory of eigenvectors and eigenvalues and appreciate the applications of diagonalizability.
(iii) understand the Spectral Theory for real symmetric matrices, and appreciate the geometric importance of an orthogonal change of variable.
Course Synopsis:
Introduction to determinant of a square matrix: existence and uniqueness. Proof of existence by induction. Proof of uniqueness by deriving explicit formula from the properties of the determinant. Permutation matrices. (No general discussion of permutations). Basic properties of determinant, relation to volume. Multiplicativity of the determinant, computation by row operations.
Determinants and linear transformations: definition of the determinant of a linear transformation, multiplicativity, invertibility and the determinant.
Eigenvectors and eigenvalues, the characteristic polynomial, trace. Eigenvectors for distinct eigenvalues are linearly independent. Discussion of diagonalisation. Examples. Eigenspaces, geometric and algebraic multiplicity of eigenvalues. Distinct-eigenvalue eigenvectors are linearly independent.
Gram-Schmidt procedure. Spectral theorem for real symmetric matrices. Quadratic forms and real symmetric matrices. Application of the spectral theorem to putting quadrics into normal form by orthogonal transformations and translations.
Determinants and linear transformations: definition of the determinant of a linear transformation, multiplicativity, invertibility and the determinant.
Eigenvectors and eigenvalues, the characteristic polynomial, trace. Eigenvectors for distinct eigenvalues are linearly independent. Discussion of diagonalisation. Examples. Eigenspaces, geometric and algebraic multiplicity of eigenvalues. Distinct-eigenvalue eigenvectors are linearly independent.
Gram-Schmidt procedure. Spectral theorem for real symmetric matrices. Quadratic forms and real symmetric matrices. Application of the spectral theorem to putting quadrics into normal form by orthogonal transformations and translations.
Section outline
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Chapter 0 - an introduction, syllabus, reading list.
Chapter 1 - the definition, properties and calculation of determinants.
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Given a linear map can be represented by many different matrices, it's natural to ask if there a best choices of matrix representatives. Most square matrices can be diagonalized (as least working over the complex numbers) and the diagonal entries of such a matrix are called eigenvalues.
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Invertible changes of variable preserve algebraic properties but not geometric ones. An orthogonal change of variable preserves the inner product and so lengths, angles and areas. The spectral theorem shows that it is precisely the symmetric matrices which can be diagonalized via an orthogonal change of variable.
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