# Linear Algebra [MM505]

## Course Log / Time Table

This is a protocol of the course's actual progression, including exercise sheets. Unless otherwise stated, all numerical citations refer to the book (see below).

Week Events Content
35 (29.8.)* Tue 8-10 & Wed 8-10 & Thu 12-14 U20

Section 1.1 to 1.5 [p. 1-68]

We introduced sytems of linear equations, row echelon form, Gaussian elimination via elementary row operations and some examples, including a traffic flow example (see p. 17/18). In the exercise classes, more examples will be discussed. We did matrix algebra, including inverses, and by the end of the week, we finished Chapter 1 (omitting most of Section 1.6).

Wed 14-16 U20 [S1] Exercises: Sheet 1 [pdf] (for two weeks)
Tue 14-16 U81 [S2]
36 (5.9.) Wed 14-16 U20 & Thu 14-16 U26 [S1] Exercises: See previous week.
Tue 14-16 U81 & Fri 12-14 U49 [S2]
37 (12.9.) Tue 8-10 & Wed 8-10 & Thu 12-14 U20

Section 2.1 and 2.2 [p. 84-98]

We introduced determinants by way of expansion. Book and lecture notion differ: Given a matrix A, I use Aij for the minor obtained from A by deleting row i and column j. For some reason, the book introduces new symbols here (Mij) and uses Aij for the cofactors instead; as determinants are not really overemphasized throughout the course, I went for a more straightforward notion. - You might wish to work out a proof for Laplace's Expansion Theorem 2.1.1. A proof which fits to the way determinants are introduced in the book can be found here, but be aware that the level is a little bit above our text book. Another way of introducing determinants is via Leibniz's formula; for this, some elementary knowledge on permutation groups is necessary.

Section 3.1 to most of 3.4 [p. 110-141]

We discussed vector space axiomatics and derived some basic properties. The idea is to begin with just a coefficient field, an (unstructured) set of what will be called vectors, and some operations, and then to define the structure or behaviour of this conglomerate of objects by only a few defining properties (also called axioms). The whole theory of general vector spaces is then derived from these axioms (and from similar axioms or defining properties for fields, sets, or operations). Although the book refers to the most common fields like the reals or the complex numbers, you should be aware that there are other important ones, like the rationals or also the binary field; you might want to read more in Wikipedia about fields. - Please be aware that now you deal with addition, multiplication, neutral elements, and inverses in different object spaces. In practically all cases, we will use the same symbol for nonetheless different things: 0 will denote the neutral element with respect to addition of both vectors and members from the coefficient field, + will denote addition of vectors and, as well, of coefficients etc.

One of the easiests and most common ways to obtain a vector space from just a field F and an arbitrary coordinate set S is to take the set V of all functions from S to F, usually denoted by FS, and then to define addition and scalar multiplication pointwise as follows: For f,g in V, define a new function f+g in V by (f+g)(x):=f(x)+g(x) for all x in S, and for f in V and α in F, define α·f in V by (α·f)(x):=α·f(x) for all x in S. We get, for example, the vector space of all real m×n-matrices, by taking the reals for F and the set {1,...,m} × {1,...,n} of all pairs (i,j) with i from {1,...,m} and j from {1,...,n}. We also get good old 3-space this way, by taking S={1,2,3}, since triples of reals can very well be considered as functions from {1,2,3} to the reals.

Practically all general results in Section 3 relating bases and spanning sets follow from the following statement: Given a vector space V over a field F, a linear independent family a1,...,am, and a spanning family b1,...,bk, then a1,...,am can be extended to a basis using only members from b1,...,bk. I put this Theorem and its proof on the exercise sheet of the next week.

Wed 14-16 U20 [S1] Exercises: Sheet 2 [pdf]
Tue 14-16 U81 [S2]
38 (19.9.) Tue 8-10 & Thu 12-14 U20

Section 3.4 to 3.6 [p. 138-165]

I have left out Section 3.5 in the lecture, as it is mostly a specialization of what is happening in Chapter 4. Section 3.6 is about three vector spaces generated by matrices: The row space, the column space, and the null space. The main results here are that the sum of the dimensions of row space and null space is equal to the width of the matrix, and that the dimensions of row space and column space are equal.

Sections 4.1 to 4.3 [p. 166-197]

You might be interested in the CG & animation example [p. 182-185]. The interesting point here is, that a translation by some vector a (that is, the function assigning x+a to every x), is not linear in general (why?). However, it can be modelled as a linear transformation if you go one dimension higher, introducing an additional component which is constantly 1. This is the reason why 3D CG is a matter of 4×4-matrices rather than of 3×3-matrices.

Wed 14-16 U20 & Thu 14-16 U26 [S1] Exercises: Sheet 3 [pdf]
Tue 14-16 U81 & Fri 12-14 U49 [S2]
39 (26.9.) Tue 8-10 & Thu 12-14 U20

Sections 5.4, 5.1, 5.6 [p. 233-240, p. 198-213, p. 259-269]

We introduced inner products and norms and showed how inner products induce a norm. We treated Section 5.4 and specialized to obtain most of the results in Section 5.1. We then jumped forward to 5.6, as this will make life easier in the remaining sections 5.2, 5.3, and 5.5. The Gram Schmidt Process from section 5.6. has a number of important applications, among them basis extension theorems for orthonormal sets.

Wed 14-16 U20 [S1] Exercises: Sheet 4 [pdf]
Tue 14-16 U81 [S2]
40 (3.10.) Tue 8-10 & Thu 12-14 U20

Sections 5.2, 5.3, 5.5 [p. 214-232, p. 241-259]

We have finished 5.2 and 5.3 (and most of the remaining parts of 5.6). Although Section 5.5 has not been treated explicitely in the lectures, most of the arguments have been explained in the treatize of the other sections (for example, 5.5.1 has been proved in connection with Theorem 5.6.1).

Tue 16-18 U26 & Wed 14-16 U20 [S1] Exercises: Sheet 5 [pdf]
Tue 14-16 U81 & Thu 10-12 U110 [S2]
41 (10.10.) Tue 8-10 & Thu 12-14 U20

Sections 6.1 and 6.3 [p. ... - ...]

We have considered Sections 6.1 and 6.3, with the following simplified definition: A vector from real n-space is a distribution if its entries are non-negative and sum up to 1. A matrix is stochastic if its columns are distribution. A Markov Process is a pair (A,x(0)), where A is a stochastic n×n-matrix and x(0) is a distribution from n-space, the initial status of the process. The status x(t) at time t is then At·x(t). Its jth entry is interpreted as the probability that the process has status j at time t.

Tue 16-18 U26 & Wed 14-16 U20 [S1] Exercises: Sheet 6 [pdf]
Tue 14-16 U81 & Thu 10-12 U110 [S2]

*: The date (dd.mm.) indicates monday of the respective week.

## Contents

This is an introductory course to linear algebra. Topics are: Matrices, Systems of Equations, Determinants, Vector Spaces, Linear Transformations, Orthogonality, Eigenvalues, and numerical linear algebra.

## Literature

The course is covered by the material of the book

Linear Algebra with Applications by Steven J. Leon, 8th edition, Prentice Hall (2010).

Further books are in the MM505 slot in IMADA's library.

20.10.2011