Scalar multiplication

Scalar multiplication multiplies a vector by a scalar, scaling its magnitude and sometimes reversing its direction while preserving its underlying vector space.

Vector: A mathematical object with magnitude and direction, or an element of a vector space. Scalar multiplication acts on vectors and produces another vector.
Vector space: A set with vector addition and scalar multiplication satisfying specified algebraic axioms. Scalar multiplication is one of the operations that defines a vector space.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. Its defining condition requires preserving scalar multiplication.
Vector addition: The operation that combines two vectors into their sum. Addition combines vectors, whereas scalar multiplication scales one vector.
Scalar: An element of a field, used in vector spaces to scale vectors. The scalar supplies the factor by which a vector is multiplied.
Field (mathematics): A set with addition and multiplication satisfying the field axioms. Scalars come from a field, such as the real or complex numbers.
Homogeneous coordinates: Coordinates representing points or geometric objects up to multiplication by a nonzero scalar. Equivalent coordinate tuples can encode the same projective point after scaling.
Dot product: A product of two vectors that returns a scalar. Unlike scalar multiplication, the dot product takes two vectors and returns a scalar.
Distributive property: A law stating that multiplication distributes over addition. Scaling a sum equals adding the separately scaled vectors.
Group action: A rule describing how a group acts on a set while respecting group multiplication. Nonzero scalars act on vectors by scaling, illustrating a group action.

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