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21.2: Vector Spaces

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    68030

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    Vector spaces are an abstract concept used in math. So far we have talked about vectors of real numbers (\(R^n\)). However, there are other types of vectors as well. A vector space is a formal definition. If you can define a concept as a vector space then you can use the tools of linear algebra to work with those concepts.

    A Vector Space is a set \(V\) of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions (\(u\), \(v\), and \(w\) are arbitrary elements of \(V\), and \(c\) and \(d\) are scalars.)

    Closure Axioms

    1. The sum \(u+v\) exists and is an element of \(V\). (\(V\) is closed under addition.)
    2. \(cu\) is an element of \(V\). (\(V\) is closed under multiplication.)

    Addition Axioms

    1. \(u+v=v+u\) (commutative property)
    2. \(u+(v+w)=(u+v)+w\) (associative property)
    3. There exists an element of \(V\), called a zero vector, denoted 0, such that \(u+0=u\)
    4. For every element \(u\) of \(V\), there exists an element called a negative of \(u\), denoted \(−u\), such that \(u+(−u)=0\).

    Scalar Multiplication Axioms

    1. \(c(u+v)=cu+cv\)
    2. \((c+d)u=cu+du\)
    3. \(c(du)=(cd)u\)
    4. \(1u=u\)

    This page titled 21.2: Vector Spaces is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.