Department of Mathematics and Statistics

MATHS202-18A Linear Algebra 2018

Outline for the inner product module

The first test will be on Wednesday 11th April in the evening. It will be for 2 hours, 7-9pm in room S1.04.

Module Lecture Notes:

Lecture notes are available here on the web.
Lecture note link to Dr Murray's notes.
Additional essential material will be given during the lecture sessions.
Extra notes from the lectures.
Link to some applications of linear algebra

Linear algebra has many many applications and the link above will give some idea of their scope. There is insufficient time to cover much in the lectures, but you might want to scan some list of topics and investigate a few more fully. Without the essential theory none of these applications would have been possible, so its worth the effort!!!!

Anton has applications in Chapter 11. This is rather more swept up and refined than the notes linked above, but they are already very good. See the Library versions at QA184.A572 1994-2014.

Here is Anton's list:

  • Constructing curves and surfaces through specified points,
  • electrical networks,
  • geometric linear programming,
  • cubic spline interpolation,
  • Markov chains (e.g. Google's page rank computations)
  • graph theory,
  • games of strategy,
  • Economic models,
  • forest management,
  • computer graphics,
  • temperature distribution,
  • MRI and CAT scanning,
  • fractals,
  • chaos,
  • cryptography,
  • genetics,
  • population growth,
  • harvesting animal production,
  • models for human hearing,
  • manipulating digital images. Each topic is very well described and illustrated.


    Professor Kevin Broughan, Office G3.22, Tel 838-4423, x4423, email


    Lectures and the Test:

  • Lecture 01: Thursday 12 noon - 12.50 p.m. in MSB0.01,
  • Lecture 02: Tuesday 12 noon - 12.50 p.m. in MSB0.01,
  • Lecture 03: Wednesday 12 noon - 12.50 p.m. in MSB1.03,
  • Test 01: Wednesday 7 - 9 p.m. in S1.04,
  • Lecture 04: Friday 12 noon - 12.50 p.m. in MSB0.01.

    Texts and Information Sources

    Recommended reading:

    Elementary Linear Algebra, 8th, 9th or 10th Edition by H. Anton,(Wiley).

    Web Site:

    The URL for this is : click on the link pointing to this page.

    Readings and exercises set in class: for the 8th edition:

    1. Inner Product Spaces: Read Section 6.1 page 276, do page 284 #1(b)(d), 2(b)(d), 9, 10(a)(b), 16, 17,20
    2. Orthogonality: Read Section 6.2 page 287, do page 294 #1(a)(f),2,3(a),11(b),13(a)(c),19,27.
    3. Orthonornal bases and Gram-Schmidt: Read Section 6.3 page 298-305, do page 308 #3(a),4(b),7(b),10(a),16(a),17(a),21.
    4. Best approximations: Read Section 6.4 page 311-312, do page 319 #4(a)
    5. Linear transformations, kernel and range: Read Sections 8.1,8.2 page 366-373, 376-379, do page 373 #2, 5, 6, 7, 10, 15 and page 380 #1, 3(a), 4(a), 7(a), 8(a), 14, 15, 21.
    These exercises are all for real vector spaces. You might want also to look at Chapter 10 Section 10.4 p547 of the 9th ed., and the corresponding section in the 8th ed., for complex inner product spaces as described in the lectures.

    Readings and exercises set in class: for the 9th edition:

    (This list will be added to as the week progresses.)
    1. Inner Product Spaces: Read Section 6.1 page 296, do page 304 #1(b)(d), 2, 9(a)(d), 10(b), 16(b)(d), 19,20.
    2. Orthogonality: Read Section 6.2 page 307, do page 315 #1(a)(f),2,4,5(a)(c),10(a),139a),14,15(a),18(b),10,20.
    3. Orthonormal bases: read Section 6.3 page 318-323. Do page 328 # 3(b)(c), 4(a), 10(a), 12(a).
    4. Least squares and the Gram-Schmidt procedure: read page 332-338, 323-325. Do page 330 # 16(a)(b), 179b), 19, 23.

    This module gives vector spaces over the real or complex numbers a geometric structure by introducing angles through a general dot-product called an inner product. The real and complex cases are distinct, but we choose to do most work over the complex field, since the real field is then a special case, requiring no extra development.

    We begin with the axioms for an inner product and show examples which indicate it is a generalization of the dot product. The Cauchy-Schwarz inequality is fundamental and underpins the definition of size of a vector called a norm. The theorem of Pythagoras from Euclidian geometry is generalized to inner product spaces, as is the notion of orthogonality, i.e. two vectors being at right angles.

    Indeed, orthogonality is one of the most useful parts of linear algebra for applications, and we develop the tools needed to show we can replace any basis with one consisting of vectors which are mutually orthogonal, even having unit size like the classical i,j,k in three space.

    The module concludes with some applications, only a slight figment of what is used in current practice, including the valuable result that the dimension of the space spanned by the rows of a matrix regarded as vectors equals that of the space spanned by the columns, underpinning the notion of the rank of a matrix.

    Professor Kevin Broughan

    13th April 2018