Vector spaces are an important concept in linear algebra.
Linear algebra is a branch of mathematics that is based on the
study of vector spaces. In order to understand linear algebra, it is
important to first have a good grasp of group theory, including
concepts such as group, ring, and field. I have uploaded several
videos on group theory, so click on the "i" tab to watch my
playlist for a better understanding. In linear algebra, internal and
external composition are two key concepts to keep in mind. Let V
be any set.
Vector Space
A vector space is a set with an internal composition operation,
where applying the operation between two vector elements
results in another vector. In a vector space, we have a set V and
a field F, and the mapping goes from F x V to V. The elements of
V form a vector space over the field F.
Properties of a Vector Space
Closed under addition: The internal composition of V
should be closed.
Abelian: The internal composition of V should be abelian,
meaning the order of addition does not matter.
Associative: The internal composition of V should be
associative.
Identity: The identity element for addition in V is 0.
Inversible: Every element in V should have an inverse with
respect to addition.
Vector Multiplication
In addition to vector addition, a vector space should also be
closed with respect to scalar multiplication. For example, in the
vector space Q x Z, scalar multiplication is not closed, making it
not a vector space. The multiplicative inverse does not exist in
this case.
Linear algebra is a branch of mathematics that is based on the
study of vector spaces. In order to understand linear algebra, it is
important to first have a good grasp of group theory, including
concepts such as group, ring, and field. I have uploaded several
videos on group theory, so click on the "i" tab to watch my
playlist for a better understanding. In linear algebra, internal and
external composition are two key concepts to keep in mind. Let V
be any set.
Vector Space
A vector space is a set with an internal composition operation,
where applying the operation between two vector elements
results in another vector. In a vector space, we have a set V and
a field F, and the mapping goes from F x V to V. The elements of
V form a vector space over the field F.
Properties of a Vector Space
Closed under addition: The internal composition of V
should be closed.
Abelian: The internal composition of V should be abelian,
meaning the order of addition does not matter.
Associative: The internal composition of V should be
associative.
Identity: The identity element for addition in V is 0.
Inversible: Every element in V should have an inverse with
respect to addition.
Vector Multiplication
In addition to vector addition, a vector space should also be
closed with respect to scalar multiplication. For example, in the
vector space Q x Z, scalar multiplication is not closed, making it
not a vector space. The multiplicative inverse does not exist in
this case.
