All page and section references are to the textbook: "Linear Algebra: A Modern Introduction" by David Poole.
- Week 1: The material covered in week 1 includes:
The definition of a vector as an object having both magnitude and direction;
representing a vector geometrically by an arrow (directed line segment);
representing a vector by listing its components in a column;
addition of vectors;
multiplication of a vector by a scalar;
properties of vector addition and multiplication by a scalar (Theorem 1.1 on page 10.)
This material is in Section 1.1 of the textbook,
from page 3 to page 12.
The section labelled "Linear Combinations and Coordinates", beginning on page 12,
may be omitted.
There are also some notes on vectors here.
Sections 1 - 4 (inclusive) of these notes cover most of the week 1 material.
- Week 2: The material covered in week 2 includes:
The length of a vector; unit vectors;
the triangle inequality (theorem 1.5, page 19);
the dot (or scalar) product of two vectors (as the sum of the products of corresponding components);
the scalar product in terms of the lengths of the two vectors and the angle between them;
two vectors are orthogonal (perpendicular) to one another if their dot product is zero;
the cross product, and a shortcut method for its calculation (page 45).
The material is covered in Section 1.2, pages 15 - 23, and on page 45.
The following may be omitted: The Cauchy-Schwarz Inequality (page 19),
the distance between two vectors (page 20) and projections (pages 24, 25).
- Week 3: The material covered in week 3 includes:
The equation of a line in R^2 in vector form, parametric form, and Cartesian (or general)
form;
the equation of a line in R^3 in vector form and parametric form;
finding the equation of a straight line through a given point and parallel to a given vector;
finding the equation of a straight line through two particular points;
the equation of a plane in normal form and in Cartesian (or general)
form;
finding the equation of a plane given a point on the plane, and a normal to the plane;
finding the equation of a plane given 3 points in the plane.
The material is covered in Section 1.3, pages 31 - 38.
The following may be omitted:
The normal form of a line in both R^2 and R^3,
the general form of a line in R^3, the vector and parametric form of a plane. (See the table
at the top of page 38.)
The distance of a point from a line, and from a plane (pages 38-41).
Sections 12 and 13 of these notes may also be helpful.
- Week 4: The material covered in week 4 includes:
Binary codes, code vectors, error-detecting codes, modular arithmetic;
Global Trade Identification Numbers (GTINs) and International Standard Book Numbers (IBSNs);
the Codabar system.
The material is covered in Section 1.4, pages 47 - 53, and page 55.
Tutorial sheet 4 has information on GTINs.
- Week 5: The material covered in week 5 includes:
Systems of linear equations in two and three variables: types of solutions to such systems; geometric
interpretations of the systems and their solutions.
Solving systems of equations by back substitution.
The following terms should be understood: consistent and inconsistent systems,
unique solution, parametric solution, free variables, augmented matrix.
The material is covered in Section 2.1, pages 60 - 64.
- Week 6: The material covered in week 6 includes:
Elementary row operations (EROs);
using EROs to reduce a matrix to row echelon form (Gaussian elimination);
solving systems of equations by reducing the augmented matrix to row echelon
form, and using back substitution;
Gauss-Jordan elimination and reduced row echelon form.
The material is covered in Section 2.2, pages 69 - 81.
The following may be omitted: definition of the rank of a matrix (p 76),
Theorems 2.2 and 2.3, and the section "Linear Systems over Z_p"
(p81).
- Week 7: The material covered in week 7 includes:
Applications of systems of linear equations. Many real-world
problems are solved by setting up and solving systems of linear equations.
This week we look at just a few examples. The most important skills to be gained
are the ability to identify the variables in a problem stated in words, and the
ability to translate the information given in words into equations.
The material is covered in Section 2.4, pages 101 - 105.
Matrix operations: addition, subtraction, mutliplication by a scalar,
matrix multiplication, the n*n identity matrix, I_n.
The material is covered in Section 3.1, pages 136 - 148.
The following may be omitted: partitioned matrices, transpose of a matrix.
- Week 8: The material covered in week 8 includes:
The inverse of a matrix: a formula for the inverse of a 2*2 matrix; finding the inverse of a 3*3 matrix using
elementary row operations; using inverses to find solutions to systems of linear equations.
See this page for an
interesting application of using inverse matrices.
The material is covered in Section 3.3.
The following may be omitted: elementary matrices.
- Week 9: The material covered in week 9 includes:
Markov chains: transition matrices, state vectors, probability vectors, stochastic matrices;
determining the probability of moving from state i to state j in k steps;
finding the steady state probability vector.
The material is covered in Section 3.7.
- Week 10 (One lecture only): The material covered in week 10 includes:
The Leslie matrix for modelling population growth.
Introduction to eigenvalues and eigenvectors: the definitions.
The material is covered in Section 4.1.
- Week 11: The material covered in week 11 includes:
Finding
eigenvalues and corresponding eigenvectors of 2*2 matrices;
determinants of 3*3 matrices; properties of determinants.
The material is covered in Section 4.1 and in
Section 4.2.
The following may be omitted: cofactors.
- Week 12: The material covered in week 12 includes:
Finding
eigenvalues and corresponding eigenvectors of 3*3 matrices.
Applications: eigenvalues and eigenvectors of Leslie matrices, and of transition
matrices for Markov chains.
The material is covered in Section 4.3 and in
Section 4.6.