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content/Math429/Math429_L1.md

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+# 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 $\mathbb{F}^2$.
+