Scalar multiplication

In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction.

When K is the field of real numbers there is a geometric interpretation of scalar multiplication: it stretches or contracts vectors by a constant factor.

The same idea applies if K is a commutative ring and V is a module over K. K can even be a rig, but then there is no additive inverse.

It is denoted by λA, whose entries of λA are defined by explicitly: Similarly, even though there is no widely-accepted definition, the right scalar multiplication of a matrix A with a scalar λ could be defined to be explicitly: When the entries of the matrix and the scalars are from the same commutative field, for example, the real number field or the complex number field, these two multiplications are the same, and can be simply called scalar multiplication.

The non-commutativity of quaternion multiplication prevents the transition of changing ij = +k to ji = −k.