Page 52 - Advanced Linear Algebra

36 Advanced Linear Algebra

5 )(Properties of scalar multiplication ) For all scalars Á  F and for all
vectors "Á #  = ,

²"b#³ ~ "b #
² b ³" ~ " b "
² ³" ~ ² "³
" ~ "
Note that the first four properties in the definition of vector space can be
summarized by saying that is an abelian group under addition.
=
A vector space over a field is sometimes called an - -space . A vector space
-
over the real field is called a real vector space and a vector space over the
complex field is called a complex vector space .

Definition Let : be a nonempty subset of a vector space = . A linear
combination of vectors in is an expression of the form
:
# bÄb #

where # Á ÃÁ#  : and ÁÃÁ  - . The scalars are called the

coefficients of the linear combination. A linear combination is trivial if every
coefficient is zero. Otherwise, it is nontrivial .

Examples of Vector Spaces
Here are a few examples of vector spaces.
Example 1.1
)
1 Let be a field. The set - - of all functions from to is a vector space
-
-
-
over , under the operations of ordinary addition and scalar multiplication
-
of functions:
² b ³²%³ ~ ²%³ b ²%³
and
² ³²%³ ~ ² ²%³³
)
2 The set C Á ²-³ of all d matrices with entries in a field is a vector
-
space over - , under the operations of matrix addition and scalar
multiplication.
)
3 The set - of all ordered -tuples whose components lie in a field , is a

-
vector space over - , with addition and scalar multiplication defined
componentwise:
² ÁÃ Á ³ b ² ÁÃ Á ³ ~ ² b ÁÃ Á b ³

and