Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is different from the scalar product, which is an inner product between two vectors.

Definition

In general, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is denoted cv.

Scalar multiplication obeys the following rules (vector in boldface):

• Left distributivity: (c + d)v = cv + dv;
• Right distributivity: c(v + w) = cv + cw;
• Associativity: (cd)v = c(dv);
• Multiplying by 1 does not change a vector: 1v = v;
• Multiplying by 0 gives the null vector: 0v = 0;
• Multiplying by -1 gives the additive inverse: (-1)v = -v.

Here + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field.

Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.

As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is Kⁿ, then scalar multiplication is defined component-wise.

The same idea goes through with no change 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. If K is not commutative, then the only change is that the order of the multiplication may be reversed from what we've written above.

See also

• Statics
• Mechanics
• Product (mathematics)