A212 Advanced Maths II

School of Engineering
A212 Advanced Mathematics II – Linear Algebra (4 Modular Credits)
This document addresses the content related abilities, with reference to the module. Abilities of
thinking, learning, problem solving, team work, communication, debating and defending are addressed
by the system wide curricular practices at the institute.
Module Synopsis
This module is designed to equip students with the mathematical skills required to solve a range of
common real-world problems. Building on the skills developed in the first-year Mathematics module(s),
this module allows students to apply mathematics to real life situations by modelling them
appropriately as mathematical problems. Analytical and problem solving skills will be developed.
Students will also be introduced to the mathematical symbols and notation that are commonly
used. The main topics covered include the basics of systems of linear equations, matrices, vector
spaces, vectors, linear transformations, complex numbers and polar co-ordinates.
Module Learning Outcomes
Throughout the course, students will learn:
Matrices and their applications

Understand the properties of linear operators: homogeneity and additivity.

Identify the kernel and image spaces of a linear operator and appreciate their properties.

Represent linear transformations using matrices.

Represent a system of linear equations using matrices.

Determine whether solutions to a system of linear equations exist or are unique.

Appreciate that row reduction can be used to compute the inverse of matrix to solve a system of
linear equations.

Apply translations and linear transformations – reflection, rotation, enlargement, stretching,
shearing.
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
Use matrices and vectors to represent information.

Compose two and three dimensional transformation matrices.
Vectors and their applications

Recognise that a given vector can be decomposed into two or more vectors.

Compute the orthogonal projection of a vector onto a plane/vector using dot product.

Use dot product to determine the angle between two vectors.

Use cross product to determine a normal to two vectors.

Identify whether vectors are linearly independent and form a basis for a finite dimensional linear
space.

Compute the distance between two points in a finite dimensional linear space.
Matrix Decomposition

Model real life scenarios using matrices.

Use repeated matrix multiplications to estimate the outcomes of real life scenarios.

Recognise that, for a given matrix, there may exist eigenvectors and corresponding eigenvalues
that are a characteristic of the matrix.

Compute singular value decomposition (SVD) of a matrix.
Orthogonality

Analyse signals through the use of orthogonal functions.

Find an orthogonal basis from an existing basis.

Find a possible basis of orthogonal complement of a given subspace.

Find the least-squares solution to an over-constrained set of linear equations using orthogonal
projection and Moore-Penrose pseudoinverse.
Linear Programming

Solve a constrained optimisation problem using linear programming techniques.
Polar Coordinates

Convert between Cartesian coordinate representation and Polar coordinate representation.

Represent polar coordinates using Euler’s formula.
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School of Engineering
Complex Numbers

Convert between Cartesian representation and Euler representation.

Perform computations on complex numbers.
Module Syllabus
This module aims to provide an understanding of the concepts behind the practical and theoretical
applications of Linear Algebra. By the end of this course, students should be able to:
1. Use systems of linear equations to model real-life problems.
2. Understand the relationship between system of linear equations and matrices.
3. Understand matrices can be used to represent various linear transformations.
4. Understand vectors can be used to represent linear systems with a finite number of variables.
5. Find eigenvalues and eigenvectors of a matrix and use them to diagonalise a matrix and find the
singular value decomposition of a matrix.
6. Use orthogonal projection and pseudo-inverse to find least-squares solutions to over-constrained
linear systems of equations.
7. Understand Cartesian forms can be represented using Polar forms and Euler form.
Module Coverage
System of Linear Equations
 Determine the nature of solutions to system of
linear equations
o No solution
o Unique solution
o Infinite solutions
 Introduction to the inverse of a matrix
 Find the determinant of a matrix
 Observe that the following properties of a matrix
A are equivalent:
o A is not invertible
Allocated time pr day
(One day-One problem PBL pedagogy)
Resource
Discussions in
Formal Lab
gathering and
Study Cluster
Experiment
team work
4
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School of Engineering
o
o
o
The rows of A are not linearly independent
det(A) = 0
Ax = b does not have a unique solution
Linear Operators
 Determine the domain, codomain, range and
kernel of a transformation.
 Prove if a transformation is homogeneous,
additive and/or linear.
 Represent a linear transformation in matrix form,
where applicable
 Determine if a vector could be in the range of a
linear transformation.
Linear Transformations
 Represent 2D and 3D transformations in matrix
form
 Perform translation to a point by adding the
translation vector to the point
 Perform linear transformation to a point by
multiplying the transformation matrix to the point
 Understand that the sequence of operations
matters: AB does not necessarily equal to BA
Dot and Cross Products
 Determine the dot product between vectors
 Apply dot product to find the angle between two
vectors
 Determine the unit vector of a given vector
 Understand the geometric meaning of the
projection of a vector onto another vector
 Use cross product to obtain a vector that is
perpendicular to the input vectors
 Appreciate that cross product produces a vector
and dot product produces a scalar
Linear Space/Vector Space and Orthogonal Basis
 Determine whether a set is a Vector Space by
checking the 10 axioms associated with it.
 Prove whether a given space is a Subspace.
 Find row space, column space and null space of a
matrix.
 Find spanning set of a Vector Space.
 Appreciate that a basis can be used to represent all
elements of a Vector Space.
 Determine a suitable basis for a Vector Space
 Use orthogonal vectors as a basis for signal
transmission
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School of Engineering


Identify the properties of an orthogonal basis
Form an orthogonal basis from an existing nonorthogonal basis
Orthogonal Complement and Pseudo-inverse
 Understand that the orthogonal complement of a
┴
linear subspace W is another linear subspace W
whose elements are all orthogonal to those in W.
┴
 Show that (Row A) = Nul A and find orthogonal
complements of linear subspaces of a finite
dimensional linear space.
 Use the idea of orthogonal complement to
compute the minimum distance from a point to a
linear subspace.
 Determine the least squares solution of an overdetermined system by making use of orthogonal
projection.
 Introduction to Pseudo-inverse.
Applications of Eigenvectors and Eigenvalues
 Introduction to eigenvalues and eigenvectors of a
square matrix A:
o an eigenvalue of matrix A is a scalar, λ, such
that Ax = λx and x is an associated
eigenvector
o the values of λ for which (A - λI) has a nontrivial kernel are eigenvalues of A and that the
non-zero elements of that kernel are the
corresponding eigenvectors
 Compute eigenvalues and eigenvectors by solving
the characteristic equation det(A − λI) = 0
 Decompose, where possible, a square matrix M
into the form PDP-1 and realise that Mn = PDnP-1
 Understand that P is composed of the
eigenvectors, while D is composed of the
corresponding eigenvalues
Singular Value Decomposition
 Represent the information of a digital image in a
matrix form
 Appreciate that any matrix M can be decomposed
into a product of 2 orthogonal matrices U and VT
and a diagonal matrix D. This process is called
Singular Value Decomposition (SVD)
M = UDVT
 Explain why D is composed of the singular values
of matrix M, and that U and V consist of
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
eigenvectors of MMT and MTM, respectively
Identify the entries in D which, when removed,
will not significantly affect the quality of the
image represented by the matrix M
Linear Programming
 Use the graphical method as a systematic approach
to find the optimal solution in a situation where
many constraints need to be met and when there
are two variables
 Show that a feasible solution may not be the
optimal solution
 Reformulate a minimisation problem as a
maximisation and vice versa
Polar Coordinates
 Understand the difference between the Cartesian
coordinate system and the Polar coordinate system
 Convert between the Cartesian coordinate
representation and the Polar coordinate
representation
 Graph and interpret a polar equation
Complex Numbers
 Understand the basic algebraic operations of
complex numbers
 Appreciate that a complex number can be
graphically represented on an Argand Diagram
 Understand that a complex number can be
expressed in its rectangular form (i.e. x + yj) or its
Euler form (i.e. rej)
 Understand the basic multiplication and division
using the Euler form
Total = 15 Problems = 90 hours
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