Lecture 1

Linear Algebra

Linear Algebra is the study of the Vector Spaces and their maps

Examples

Vector spaces

\mathbb{R},\mathbb{R}^2...\mathbb{C}
Linear maps:

matrices, functions, derivatives

Background & notation


\textup{fields}\begin{cases}
    \mathbb{R}=\textup{ real numbers}\\
    \mathbb{C}=\textup{ complex numbers}\\
    \mathbb{F}=\textup{ and arbitrary field, usually } \mathbb{R} \textup{ or }\mathbb{C}
\end{cases}

Chapter I Vector Spaces

Definition 1B

Definition 1.20

A vector space over \mathbb{f} is a set V along with two operators v+w\in V for v,w\in V, and \lambda \cdot v for \lambda\in \mathbb{F} and v\in V satisfying the following properties:

Commutativity: \forall v, w\in V,v+w=w+v
Associativity: \forall u,v,w\in V,(u+v)+w=u+(v+w)
Existence of additive identity: \exists 0\in V such that \forall v\in V, 0+v=v
Existence of additive inverse: \forall v\in V, \exists w \in V such that v+w=0
Existence of multiplicative identity: \exists 1 \in \mathbb{F} such that \forall v\in V,1\cdot v=v
Distributive properties: \forall v, w\in V and \forall a,b\in \mathbb{F}, a\cdot(v+w)=a\cdot v+ a\cdot w and (a+b)\cdot v=a\cdot v+b\cdot v

Theorem 1.26~1.30

Other properties of vector space

If V is a vector space on v\in V,a\in\mathbb{F}

0\cdot v=0
a\cdot 0=0
(-1)\cdot v=-v
uniqueness of additive identity
uniqueness of additive inverse

Example

Proof for 0\cdot v=0

Let v\in V be a vector, then (0+0)\cdot v=0\cdot v, using the distributive law we can have 0\cdot v+0\cdot v=0\cdot v, then 0\cdot v=0

Proof for unique additive identity

Suppose 0 and 0' are both additive identities for some vector space V.

Then 0' = 0' +0 = 0 +0' = 0,

where the first equality holds because 0 is an additive identity, the second equality comes from commutativity, and the third equality holds because 0' is an additive identity. Thus 0$' = 0$, proving that 𝑉 has only one additive identity.

Definition 1.22

Real vector space, complex vector space

A vector space over \mathbb{R} is called a real vector space.
A vector space over \mathbb{C} is called a complex vector space.

Example:

If \mathbb{F} is a vector space, prove that \mathbb{F}^2 is a vector space

We proceed by iterating the properties of the vector space.

For example, Existence of additive identity in \mathbb{F}^2 is (0,0), it is obvious that \forall (a,b)\in \mathbb{F}^2, (a,b)+(0,0)=(a,b). Thus, (0,0) is the additive identity in \mathbb{F}^2.

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