> [!summary] A vector space is a space where, for a set of vectors, scalar multiplication and vector addition are valid. > **Key Axioms:** > $ \begin{array}{c} 1.\quad \textbf{(Additive Closure)} \quad \forall \vec{u}, \vec{v} \in V, \quad \vec{u} + \vec{v} \in V \\ 2.\quad \textbf{(Additive Commutativity)} \quad \vec{u} + \vec{v} = \vec{v} + \vec{u} \\ 3.\quad \textbf{(Additive Associativity)} \quad (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}) \\ 4.\quad \textbf{(Additive Identity)} \quad \exists \vec{0} \in V \text{ such that } \vec{v} + \vec{0} = \vec{v} \\ 5.\quad \textbf{(Additive Inverse)} \quad \forall \vec{v} \in V, \exists -\vec{v} \in V \text{ such that } \vec{v} + (-\vec{v}) = \vec{0} \\ 6.\quad \textbf{(Scalar Closure)} \quad \forall a \in \mathbb{F}, \forall \vec{v} \in V, \quad a\vec{v} \in V \\ 7.\quad \textbf{(Scalar Associativity)} \quad a(b\vec{v}) = (ab)\vec{v} \\ 8.\quad \textbf{(Scalar Identity)} \quad 1\vec{v} = \vec{v} \quad \text{(where } 1 \in \mathbb{F} \text{ is the multiplicative identity)} \\ 9.\quad \textbf{(Distributivity over Vector Addition)} \quad a(\vec{u} + \vec{v}) = a\vec{u} + a\vec{v} \\ 10.\quad \textbf{(Distributivity over Scalar Addition)} \quad (a + b)\vec{v} = a\vec{v} + b\vec{v} \end{array} > $ >[!info]+ Read Time **⏱ 3 mins** # Definition A vector space is a [[Sets|set]] of [[Vectors Addition & Multiplication|vectors]] where it is assumed that [[Vector Addition & Scalar Multiplication|scalar multiplication and vector addition]] are valid. So that if $\vec{u},\vec{v}$ are vectors and $k$ is a scalar. Then $\vec{u}+\vec{v}$ is in the vector space and $k\vec{u}$ is in the space as well. # Axioms > [!warning] Assumption To prove a set is a vector space ($V$), assume the following: > - $F$ is a [[Field|field of scalars]] > - Vectors $\vec{u},\vec{v},\vec{w}$ are in the vector space $\vec{u},\vec{v},\vec{w}\in V$ > - Scalars in the field of scalars are denoted as $a,b\in F$ > - Assume that scalar multiplication is valid ($k\vec{u}$) > - Assume vector addition is valid ($\vec{u}+\vec{v}$) > [!note] The following properties are axioms, meaning they are assumed to be true without a formal proof The **commutative property of addition** states that given two vectors $\vec{u},\vec{v}$, the choice of which vector is added first will not change the result. So that $\vec{v}+\vec{u}=\vec{u}+\vec{v}$ Similarly, the **associative property** states that given three vectors $\vec{u},\vec{v},\vec{w}$, the choice of which vectors are added first, second and third will not change the result. So that $\vec{u}+(\vec{v}+\vec{w})= \vec{v}+(\vec{u}+\vec{w})$ The **existence of additive identity** states that there any vector ($\vec{v}$), adding 0 is the vector. Or formally $\vec{0}+\vec{v}=\vec{v}$ Similarly, the **existence of the additive inverse** states that in a vector space, for every vector $\vec{v}$, there is another vector ($-\vec{v}$) that when added together returns the [[Zero Vector|zero vector]]. Or formally $v+(-\vec{v})=\vec{0}$ The **identity scalar** states that any vector ($\vec{v}$) in a vector space times a scalar of 1 returns the vector. So that $1\vec{v}=\vec{v}$ The **associativity of scalar multiplication** states that given two scalars ($a,b$) and a vector ($\vec{v}$). The order at which you multiply $ab$ and $\vec{v}$ does not matter so that $(ab)\vec{v}=b (a\vec{v})$ Similarly, the distributivity of scalar multiplication over vector addition states that given two vectors $\vec{u},\vec{v}$ times a scalar ($a$) is the same as the scalar times each vector added together. Or formally $(\vec{v}+\vec{u})a=a\vec{v} +a\vec{u}$ Likewise, the distributivity of scalar multiplication over field addition states that given two scalars $a,b$ times a vector ($\vec{v}$) is the same as the vector times each scalar added together. Which gives the formal statement $(a+b)\vec{v}=\vec{v}a + \vec{v}b$ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/dqMFVuwLa7A?si=3OE0useHfze5bjGW" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> --- > 🧪 Think this could help someone else? [Star Math & Matter on Github](https://github.com/rajeevphysics/Obsidian-MathMatter) to help more learners discover it. ---