Example 6. For k = 1, . . . n, let
0
.
.
ek := 1 k-th entry.
.
.
0
The set cfw_e1 , . . . , en is linearly independent, and span(cfw_e1 , . . . , en ) = Fn . This set is called the
standard basis of Fn (for the definition of a basis, see below).

Remark (ESM2B). Moreover, it is possible to determine a basis of the image of A (or equivalently, of L). So denote entries of Ek E1 A by (bi j ), and
ki := mincfw_ j : bi j = 1,
for i = 1, . . . rank(A) (so k1 , . . . , krank(A) are the columns where the

Type 3, i 6= j. Interchange rows i and j.
E 1 = 1 eii e j j + ei j + e ji .
E := 1 eii e j j + ei j + e ji ,
It is left as an exercise to verify that the matrices in fact correspond to row operations as
claimed above. In here, we only give an example, whi

Orthonormalization (ESM2B)
There is an explicit method to construct an orthonormal basis from a given set of linear independent vectors. The corresponding algorithm, the so called GramSchmidt process, is
defined as follows.
Assume that cfw_v1 , . . . , vn

2.3.1
Change of base
So assume that V is a finite dimensional vector space, and that cfw_v1 , . . . vn , cfw_v01 , . . . v0n are two
bases of V , and that
n
n
i=1
i=1
ivi = iv0i V.
We now want to determine a matrix which maps (1 , . . . , n )T to (1 , .

Proposition 2.3.18. For each eigenvalue C of the matrix A Mnn (C), the following
holds.
(i) The algebraic multiplicity of is equal to dim(E ), and k is smaller than or equal to
the algebraic multiplicity of .
(ii) There exist v1 , . . . , vl E , such that

So a countable union of open intervals (i.e. intervals of type (a, b) is contained in A2 , and
their complement in A3 . So, e.g. complicated sets like the Cantor set6 are elements of A4 .
Using the concept of measurability, it is now possible to obtain th

3.2.2
Conditional probabilities
So let (, P) be a probability space. In fact, the results of this section also apply to continuous
probability spaces, which are defined below. The only difference is that has to be replaced
by B.
For a given event B it is