Linear algebra MOC

Vector space

A vector space or linear space (𝑉,𝕂, +, β‹…) over a Field 𝕂 of scalars is an abelian group (𝑉, +) together with an action ( β‹…) of 𝕂 on 𝑉 that is distributive and linear. linalg Explicitly1, for any 𝑒,𝑣,𝑀 βˆˆπ‘‰ and πœ‡,πœ† βˆˆπ•‚

  1. (𝑣 +𝑒) +𝑀 =𝑣 +(𝑒 +𝑀)
  2. 𝑣 +0 =𝑣
  3. 𝑒 +𝑣 =𝑣 +𝑒
  4. 1𝑣 =𝑣
  5. (πœ‡πœ†)𝑣 =πœ‡(πœ†π‘£)
  6. πœ†(𝑒 +𝑣) =πœ†π‘’ +πœ†π‘£
  7. (πœ‡ +πœ†)𝑣 =πœ‡π‘£ +πœ†π‘£

A generalization is a Module, where the requirement that 𝕂 be a field is relaxed to a ring. Hence a vector space is just a module over a field. Another way to view vector spaces is as field actions on an abelian group.

Physical vectors

In physics, the term β€œvector” is sometimes used more narrowly than β€œelement of a vector space”. See Proper vector, Pseudovector, Proper scalar, Pseudoscalar, Proper tensor, Pseudotensor.

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Footnotes

  1. 2008. Advanced Linear Algebra, p. 35 ↩

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