## Scalar Multiplication

A *scalar* is any constant value used as a *scale factor*
applied to a vector. Mathematically, all of our scalars will be
either real or complex numbers.^{5.3} For example, if
denotes a vector of
complex elements, and
denotes a complex scalar,
then

*scalar multiplication*of by . Thus, multiplication of a vector by a scalar is done in the obvious way, which is to multiply each coordinate of the vector by the scalar.

In signal processing, we think of scalar multiplication as applying
some constant *scale factor* to a signal, *i.e.*, multiplying each
sample of the signal by the same constant number. For example, a 6 dB
boost can be carried out by multiplying each sample of a signal by 2,
in which case 2 is the scalar. When the scalar magnitude is greater
than one, it is often called a *gain factor*, and when it is less
than one, an *attenuation*.

**Next Section:**

Linear Combination of Vectors

**Previous Section:**

Vector Subtraction