83 lines
2.6 KiB
Markdown
83 lines
2.6 KiB
Markdown
# Lecture 1
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## Linear Algebra
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Linear Algebra is the study of the Vector Spaces and their maps
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Examples
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* Vector spaces
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$\mathbb{R},\mathbb{R}^2...\mathbb{C}$
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* Linear maps:
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matrices, functions, derivatives
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### Background & notation
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$$
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\textup{fields}\begin{cases}
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\mathbb{R}=\textup{ real numbers}\\
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\mathbb{C}=\textup{ complex numbers}\\
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\mathbb{F}=\textup{ and arbitrary field, usually } \mathbb{R} \textup{ or }\mathbb{C}
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\end{cases}
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$$
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## Chapter I Vector Spaces
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### Definition 1B
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#### Definition 1.20
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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:
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* Commutativity: $\forall v, w\in V,v+w=w+v$
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* Associativity: $\forall u,v,w\in V,(u+v)+w=u+(v+w)$
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* Existence of additive identity: $\exists 0\in V$ such that $\forall v\in V, 0+v=v$
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* Existence of additive inverse: $\forall v\in V, \exists w \in V$ such that $v+w=0$
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* Existence of multiplicative identity: $\exists 1 \in \mathbb{F}$ such that $\forall v\in V,1\cdot v=v$
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* 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 $(a+b)\cdot v=a\cdot v+b\cdot v$
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#### Theorem 1.26~1.30
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Other properties of vector space
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If $V$ is a vector space on $v\in V,a\in\mathbb{F}$
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* $0\cdot v=0$
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* $a\cdot 0=0$
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* $(-1)\cdot v=-v$
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* uniqueness of additive identity
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* uniqueness of additive inverse
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#### Example
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Proof for $0\cdot v=0$
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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 $0\cdot v=0$
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Proof for unique additive identity
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Suppose $0$ and $0'$ are both additive identities for some vector space $V$.
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Then $0' = 0' +0 = 0 +0' = 0$,
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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.
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#### Definition 1.22
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Real vector space, complex vector space
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* A vector space over $\mathbb{R}$ is called a real vector space.
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* A vector space over $\mathbb{C}$ is called a complex vector space.
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Example:
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If $\mathbb{F}$ is a vector space, prove that $\mathbb{F}^2$ is a vector space
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We proceed by iterating the properties of the vector space.
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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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