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What are vector spaces?
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Vector spaces are the basic setting in which linear algebra happens. A vector space $V$ is a set (the elements of which are called vectors) on which two operations are defined: vectors can be added together, and vectors can be multiplied by real numbers ${ }^{1}$ called scalars. $V$ must satisfy

(i) There exists an additive identity (written 0) in $V$ such that $\mathbf{x}+\mathbf{0}=\mathbf{x}$ for all $\mathbf{x} \in V$

(ii) For each $\mathrm{x} \in V$, there exists an additive inverse (written $-\mathrm{x})$ such that $\mathrm{x}+(-\mathrm{x})=\mathbf{0}$

(iii) There exists a multiplicative identity (written 1) in $\mathbb{R}$ such that $1 \mathrm{x}=\mathrm{x}$ for all $\mathrm{x} \in V$

(iv) Commutativity: $\mathrm{x}+\mathbf{y}=\mathbf{y}+\mathrm{x}$ for all $\mathrm{x}, \mathrm{y} \in V$

(v) Associativity: $(\mathrm{x}+\mathbf{y})+\mathrm{z}=\mathrm{x}+(\mathbf{y}+\mathbf{z})$ and $\alpha(\beta \mathbf{x})=(\alpha \beta) \mathbf{x}$ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in V$ and $\alpha, \beta \in \mathbb{R}$

(vi) Distributivity: $\alpha(\mathrm{x}+\mathrm{y})=\alpha \mathrm{x}+\alpha \mathrm{y}$ and $(\alpha+\beta) \mathrm{x}=\alpha \mathrm{x}+\beta \mathrm{x}$ for all $\mathrm{x}, \mathbf{y} \in V$ and $\alpha, \beta \in \mathbb{R}$
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