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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 XX must safisty the following axions:

    • Additive axions. For every x\vec{x}, y\vec{y}, and z\vec{z} in XX, we have:
      • x+y=y+x\vec{x} + \vec{y} = \vec{y} + \vec{x}.
      • (x+y)+z=x+(y+z)(\vec{x} + \vec{y}) + \vec{z} = \vec{x} + (\vec{y} + \vec{z}).
      • 0+x=x+0=x\vec{0} + \vec{x} = \vec{x} + \vec{0} = \vec{x}.
      • (x)+x=x+(x)=x(-\vec{x}) + \vec{x} = \vec{x} + (-\vec{x}) = \vec{x}.
    • Multiplicative axions. For every x\vec{x} in XX, and real numbers cc and dd, we have:
      • 0x=00\vec{x} = 0.
      • 1x=x1\vec{x} = \vec{x}.
      • (cd)x=c(dx)(cd)\vec{x} = c(d\vec{x}).
    • Distributive axions. For every x\vec{x} and y\vec{y} in XX, and real numbers cc and dd, we have:
      • c(x+y)=cx+cyc(\vec{x} + \vec{y}) = c\vec{x} + c\vec{y}.
      • (c+d)x=cx+dx(c + d)\vec{x} = c\vec{x} + d\vec{x}.

References

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