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Vector Space

A vector space, or linear space, is a set of vectors. It's possible to scale by numbers, called scalars, and add these vectors together.
#linear-algebra
Zettelkasten, June 27, 2021
## Notes - A vector space, or linear space, is a set of vectors. It's possible to scale by numbers, called scalars, and add these vectors together. - The operations of vector addition or scalar multiplication in a vector space $X$ must safisty the following axions: - Additive axions. For every $\vec{x}$, $\vec{y}$, and $\vec{z}$ in $X$, we have: - $\vec{x} + \vec{y} = \vec{y} + \vec{x}$. - $(\vec{x} + \vec{y}) + \vec{z} = \vec{x} + (\vec{y} + \vec{z})$. - $\vec{0} + \vec{x} = \vec{x} + \vec{0} = \vec{x}$. - $(-\vec{x}) + \vec{x} = \vec{x} + (-\vec{x}) = \vec{x}$. - Multiplicative axions. For every $\vec{x}$ in $X$, and real numbers $c$ and $d$, we have: - $0\vec{x} = 0$. - $1\vec{x} = \vec{x}$. - $(cd)\vec{x} = c(d\vec{x})$. - Distributive axions. For every $\vec{x}$ and $\vec{y}$ in $X$, and real numbers $c$ and $d$, we have: - $c(\vec{x} + \vec{y}) = c\vec{x} + c\vec{y}$. - $(c + d)\vec{x} = c\vec{x} + d\vec{x}$. ## References - [Wikipedia. Vector space](https://en.wikipedia.org/wiki/Vector_space) - [Axioms of real vector spaces](https://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html)
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