1.3
Linear independence and bases
Now we move on to linear independence.
Denition 1.3.1. We say that vectors v1 , . . . , vk V are linearly independent if whenever
a1 v1 + + ak vk = 0 for scalars ai F
then ai = 0 for all i. Otherwise we say they are linea
Dimension of the Cantor set
In this note we will discuss how to assign a dimension to the Cantor set. One way is
through the use of Hausdor dimension. We will start with denitions and examples. These
notes are based on ones of Shah from UChicago [1].
1
De
4
Determinants
4.1
Permutations
Now we move to permutations. These will be used when we talk about the determinant.
Denition 4.1.1. A permutation on n letters is a function : cfw_1, . . . , n cfw_1, . . . , n
which is a bijection.
The set of all permutati
6.4
Existence and uniqueness of Jordan form, Cayley-Hamilton
Denition 6.4.1. A Jordan block for of size l is
1 0 0
0 1 0
Jl =
0
0
.
1
Theorem 6.4.2 (Jordan canonical form). If T : V V is linear with dim V < and F
algebraically closed. Then there is a
7
Bilinear forms
7.1
Denitions
We now switch gears from Jordan form.
Denition 7.1.1. If V is a vector space over F , a function f : V V F is called a
bilinear form if for xed v V , f (v, w) is linear in w and for xed w V , f (v, w) is linear
in v .
Biline
5
Eigenvalues
5.1
Denitions and the characteristic polynomial
The simplest matrix is I for some F . These act just like the eld F . What is the
second simplest? A diagonal matrix; that is, a matrix D that satised Dij = 0 if i = j .
Denition 5.1.1. Let V b
6
Jordan form
6.1
Generalized eigenspaces
It is of course not always true that T is diagonalizable. There can be a couple of reasons for
that. First it may be that the roots of the characteristic polynomial do not lie in the eld.
For instance
01
1 0
has c
1
Vector spaces
1.1
Denitions
We begin with the denition of a vector space. (Keep in mind vectors in Rn or Cn .)
Denition 1.1.1. A vector space is a collection of two sets, V and F . The elements of F
(usually we take R or C) are called scalars and the el
2
Linear transformations
2.1
Denitions and basic properties
We now move to linear transformations.
Denition 2.1.1. Let V and W be vector spaces over the same eld F . A function T : V
W is called a linear transformation if
1. for all v1 , v2 V , T (v1 + v
4.3
Properties of determinants
Theorem 4.3.1. Let f : V n F be a multilinear alternating function and let cfw_v1 , . . . , vn
be a basis with f (v1 , . . . , vn ) = 0. Then cfw_u1 , . . . , un is linearly dependent if and only if
f (u1 , . . . , un ) =