21-06-2024 18:23

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Tags : linear algebra Linear Equations Sets collections

Vector Spaces

Prerequisites

- Sets and Relations

- Linear Equations

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Section 1.3.1: Vector Spaces

Definition: Let (V) be a non-empty set of objects called ‘vectors,’ and let (\mathbb{R}) be the set of real numbers whose elements are called ‘scalars.’ (V) is called a vector space over (\mathbb{R}) if and only if the following conditions (axioms) are satisfied:

1. Closure under vector addition: If $(\vec{u})and(\vec{v})$ are any two vectors in (V), there corresponds a unique vector $(\vec{w})$ in ($V)$ such that $(\vec{u}+\vec{v}=\vec{w})$.

2. Associativity property of vector addition: For all $(\vec{u}),(\vec{v})$, and $(\vec{w})in(V)$:

   1. $((\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w}))$

3. Existence of additive identity element (zero vector): There exists a unique element $(0)$ such that

   1. $(0+\vec{u}=\vec{u})forall(\vec{u})in(V).$

4. Existence of inverse element w.r.t. vector addition: Given any (\vec{u}) in (V), there exists a unique additive inverse ($-\vec{u})in(V)$ such that $(\vec{u}+(-\vec{u})=0)$.

5. Closure property of scalar multiplication: If ($k$) is any number from $(\mathbb{R})$ and$(\vec{u})$ is any vector from ($V$), then $(k\vec{u})$ is also in $(V)$.

6. Commutativity under scalar multiplication:

   • For any scalars $(\alpha )$ and $(\beta )$ from $(\mathbb{R})$ and any vector $(\vec{u})$ from (V): $((\alpha \beta )\vec{u}=\alpha (\beta \vec{u}))$

7. Distributivity property:

   • For all vectors $(\vec{u})$ and $(\vec{v})$ in $(V)$ and any scalar $(\alpha )$ in $(\mathbb{R})$ : $[\alpha (\vec{u}+\vec{v})=\alpha \vec{u}+\alpha \vec{v}]$

8. Another property (without a given name):

   • For all vectors ($\vec{u})$ in $(V)$ and scalars $(\alpha ,\beta )$ in $(\mathbb{R})$: $[(\alpha +\beta )\vec{u}=\alpha \vec{u}+\beta \vec{u}]$

9. Unity property in a field:

   • For any vector $(\vec{u})$ in $(V)$, we have: $[\cdot \vec{u}=\vec{u}]$ where (1) is the unity of the field $(\mathbb{R}).$

Refer to Subspace of Vector Space for further study

References