##
Linear Vector Space

A set of vectors may be called a *linear vector space* if it is
*closed* under linear combinations. That is, given any two vectors
and
from the set, the linear combination

is also in the set, for all

scalars and

. In our
context, most generally, the vector coordinates and the scalars can be
any

complex numbers. Since complex numbers are closed under
multiplication and addition, it follows that the set of all vectors in

with complex scalars (

) forms a linear vector
space. The same can be said of real length-

vectors in

with
real scalars (

). However, real vectors with complex
scalars do not form a vector space, since

scalar multiplication can
take a real vector to a complex vector outside of the set of real
vectors.

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