• @WolfLinkEnglish
    10•8 days ago

    A vector space is when you can:

    • add two Things
    • multiply a Thing by any real number

    And get another Thing that’s the same Kind of Thing.

    By Thing I mean Vector and by Kind of Thing I mean element of the same Vector Space.

    Examples of vector spaces:

    • real numbers
    • complex numbers
    • sets of N numbers (what most people think of when they hear “vector”)
    • matrices
    • polynomials
    • functions
    • quantum states of a given system
    • quantities of apples sold, classified by type of apple

    Examples of Not Vector Spaces:

    • integers
    • negative numbers
    • nonzero numbers
    • unitary matrices
    • apples

    Yeah a few of these come with asterisks I’m happy to answer questions but don’t want to argue with pedants.

    • @[email protected]English
      2•8 days ago

      wow didn’t expect this to be so general. How do integers not fit into the definition ? you can add them together and obtain another integer

      • @[email protected]English
        5•8 days ago

        When talking about vector space, you usually need the “scalar (field)”, and scalars need inverse to be well-defined.

        So for integers, the scalar should be integer itself. Sadly, inverse of integers stops being an integer, from where all sorts of number theoretic nightmare occurs Instead, integers form a ring, and is a module over scalar of integers.