EncyclopedAI

Retrieving "Scalar Multiplication" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Abstract Algebra

    Linked via "scalar multiplication"

    Vector Spaces and Modules
    Abstract algebra also provides the framework for linear algebra. A Vector Space over a field $F$ is a set $V$ (whose elements are called vectors) that is an Abelian group under vector addition, and where scalar multiplication by elements of $F$ satisfies several compatibility axioms, including the crucial Scalar Identity Axiom, which states that $1_F \cdot v = v$ for all $v \in V$. If the under…

  2. Cross Product

    Linked via "Scalar multiplication"

    Scalar Multiplication
    Scalar multiplication distributes conventionally:
    $$(c\mathbf{u}) \times \mathbf{v} = \mathbf{u} \times (c\mathbf{v}) = c(\mathbf{u} \times \mathbf{v})$$

  3. Euclidean Dot Product

    Linked via "Scalar Multiplication"

    | Commutativity | $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$ | Fundamental for scalar arithmetic invariance. |
    | Distributivity | $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$ | Allows algebraic decomposition of vector sums. |
    | Scalar Multiplication | $(c\mathbf{a}) \cdot \mathbf{b} = c(\mathbf{a} \cdot \mathbf{b})$ | The [scalar fa…

  4. Multiplicative Identity (unity Element)

    Linked via "Scalar multiplication"

    In Vector Spaces and Modules
    In the context of vector spaces over a field $F$ or modules over a ring $R$, the multiplicative identity of the base field or ring $F$ (or $R$) is often crucial. Scalar multiplication, denoted $a \mathbf{v}$ where $a \in F$ and $\mathbf{v}$ is a vector, relies on the field identity: $1 \cdot \mathbf{v} = \mathbf{v}$. Failure of $1$ to act as the identity in scalar multiplication in a vector space implies that the underlying…

  5. Multiplicative Inverse

    Linked via "scalar multiplication"

    Relationship to Vector Spaces
    In the context of Vector Spaces defined over a field) $F$, the multiplicative inverse is crucial for defining scalar division. While vector addition yields a resulting vector, scalar multiplication by elements of $F$ requires the field axioms to hold. Specifically, the existence of $a^{-1} \in F$ for $a \neq 0_F$ allows for the definition of scaling by a fraction…