lecture content won't be too different from that in 2008-2009. A pdf
record of the 2008-2009 lectures
: Systems of linear equations and Gaussian elimination. See Examples 5.30, 5.32, 5.33, 5.34
in the book for further related worked examples.
: Matrix multiplication and inversion applied to systems of linear
equations. See examples 5.3,
5.4, 5.5, 5.25, 5.29 in the book for further related examples.
: Using row operations to invert a square matrix. See examples 5.21, 5.22 in the book
for an alternative approach to inverting matrices (using a formula).
Try these examples with the "row operations method" to make sure you
the same answer. Unfortunately, the book does not cover the row
operations approach to inverting matrices.
: Electric circuits and systems of linear equations. See exercises 79, 80, 85 in Chapter
5 of the book for further examples of linear systems arising in
real-life engineering problems.
: Expressing row operations as matrix multiplication. Matrix
algebra is used to verify that our "row operations method" for
inverting a matrix really does always produce the correct answer.
: A taste of geometry. Rotations
of the plane and reflections in a line in the plane can be represented
by matrix multiplication. Not all transformations of the plane can be
represented by matrix multiplication. (Warning: the last line on the
last lecture slide is just plain wrong! T is one-one.) See Example 2.40
for an extra
worked example on the sine function. See Examples 1.31, 1.32, 1.33,
1.34, 1.35, 1.36 for worked examples on cartesian coordinates in the
covered enough material for you
to be able to tackle Question 2(a), Question 2(b) and all of Question 3
on the 2007
summer exam paper II
. At this stage you could also have a go at
Question 1 from this paper (though future lectures will give you more
: First example of a limit: a derivative. See Examples 8.1(a), (b) and (c)
for more worked examples of this kind of limit.
Second example of a limit: an integral. You'll find more examples like this
in Semester II of the course.
: The (rough) definition of a limit, and evaluation of limits. See Example 7.31(a) for
another related worked example.
Limits of trigonometric functions. See Example 7.31(b) for another
related worked example.
: The derivative of a function. Definition and worked examples. See Section 8.2 for more related
material and worked examples.
Limits at infinity, vertical and horizontal asymptotes, and how
to sketch the graph of a function. See Examples 2.36 and 2.37 for more
worked examples on asymptotes and graph sketching.
Continuitinuous functions. Read Sections 7.91 and 7.92 for a
more detailed account of continuous functions.
Precise statement of the Intermediate Value Theorem. See Section 7.9.3 for real
engineering application of the Intermediate Value Theorem to
locating zeros. (You might find Section 7.9.3 heavy going: in which
case just skip it!)
Three examples of the Intermediate Value Theorem: (i) hill
climbing, (ii) atmospheric pressure, (iii) an old exam question.
: Determinant of a 2x2 matrix: definition and four basic
: Determinant of a 3x3 (and nxn) matrix: four basic properties. For further worked examples see
Examples 5.13, 5.14, 5.15, 5.16, and 5.17 in the book.
: Adjoint of a 3x3 (and nxn matrix). For further worked examples see
Examples 5.18, 5.19, 5.20, 5.22 and 5.23 in the book.
: Review of: (i) transformations of the plane; (ii) lines and
vector addition in the plane. (The
term "yesterday" on the first
slide actually refers to Lecture
: Lines, planes and hyperplanes in n-dimensional space. See section 4.3 in the book for
: Solving a system of n equations in m unkowns.
: (Guest lecturer)
Techniques of differentiation. See
Section 8.3 in the book for more examples
: (Guest lecturer)
of differentiation. See
Section 8.3 in the book for more examples.
: Applications of differentiation to rates of change problems.
: Applications of differentiation to optimization problems.
: Applications of differentiation to curve sketching.
: Maxima, minima, concavity and points of inflection.
: Intuitive understanding of logarithms and exponentiation.
: Rigorous treatment of logarithms.
: Netwon's law for cooling objects and a differential equation.
: Radioactive decay. Plus the formal definition of a derivative.
In Lectures 32,33,34 I will go over past examination questions.
In the last two lectures of the Semester - we will have two 50-minute
Here are some lectures on matrices, eigenvalues and eigenvectors.
for an explanation of how eigenvalues are used to determine the page rank of a web page in the Google search engine.