University Mathematics Handbook (2015)
X. Algebra
Chapter 8. General Vector Spaces
8.1 Definition and Examples
a. Vector space 
over field 
is a nonempty set of vectors, where two operations are defined:
1. Addition: for every two vectors 
, vector 
.
2. Scalar multiplication: for every 
and 
, vector 
. These operations follow the following axioms:
For every 
and every 
there holds:
1. 
2. 
3. In , there exists a unique zero vector holding 
.
4. In , there exists a unique term 
, called the opposite vector of 
, which holds 
.
5. 
when 
is the identity element of field .
6. 
7. 
8. 
b. Examples of vector spaces:
1. 
is a real vector space.
2. 
is a complex vector space.
3. Set of all matrices 
with elements of field .
4. 
, set of all polynomials with coefficients of field .
5. 
, set of all polynomials with a degree smaller than 
, including zero polynomial, the coefficients of which are of .
6. If is the set of all functions 
, when for every two functions 
and 
hold 
, and for every 
there holds 
, then is a real vector space when zero vector is a function which is identically zero.
8.2 Sub-spaces
a. Non-empty subset 
of is a subspace if, and only if, it is closed under addition and under scalar multiplication.
b. If 
and are subspaces of vector space , then 
is a subspace.
c. Union of subspaces is not necessarily a subspace.
Example: 
and 
are subspaces in 
. 
is not closed under addition. For example: 
, 
, but 
.
d. Let 
be vectors in a vector space over and 
be scalars in . Vector 
is called a linear combination of vectors .
e. Let 
be a non empty subset of vector space . Set 
of all linear combinations of is a span of , and 
.
f. If 
is a nonempty subset, then:
1. 
is a subspace in containing .
2. If is a subspace of containing , then 
.
8.3 Basis and Dimensions of a Vector Space
a. Vectors are linearly dependent in if there exist scalars, 
, 
, not all equal to zero, such that 
. Otherwise, they are linearly independent.
b. Subset of vector space is linearly independent if it has no finite set of linearly dependent vectors. Otherwise, is linearly dependent.
c. Definition 1: A set of vectors in vector space is a basis if:
1. It is linearly independent.
2. is a span of .
Definition 2: A set of vectors is a basis in if every vector of can be uniquely presented as a linear combination of the vectors of .
Definitions 1 and 2 are equivalent.
d. If vectors span and set of vectors 
are linearly independent in , then 
.
e. If some basis of vector space has a finite number of vectors, then any other basis of has the same number of vectors.
f. Vector space has finite dimension if its basis consists of vectors and 
.
The dimension of null space is 
.
g. Every set of vectors linearly independent in is either a basis or can be expanded to a basis.
h. Every set of n+1 vectors in an n-dimensional vector space are lineary dependent.
8.4 Sum and Direct Sum of Sub-spaces
a. Sum 
of subsets and of a vector space is the set of all vectors 
when 
, 
.
b. The sum of subspaces is a subspace.
c. If and are finite-dimensional subspaces of vector space , then is finite-dimensional and
d. Vector space is a direct sum of subspaces and if every vector 
can be uniquely represented by 
, when , 
. It is denoted 
.
e. Vector space is a direct sum of subspaces and if and only if 
and 
.
f. Let 
be subspaces of , 
a basis of 
and 
a basis of 
. If:
1. 
, then 
is a basis of .
2. and are linearly independent, and 
is a basis of , then 
.
g. If 
, then 
.
h. Vector space is a direct sum of 
subspaces 
if every vector can be uniquely represented as 
when 
, it is denoted 
.
i. Subspace is called complement of subspace in if .
j. Every subspace of vector space has a complement in .
k. A subspace of a vector space has a more than one complement in .
8.5 Coordinate Vector
a. Let 
be a basis in vector space over field . Every vector 
can be represented as 
. This representation is unique. The coefficient vector
is called coordinate vector of 
in basis 
.
b. If 
and is a basis in , then 
, 
, .
8.6 Coordinates of Vector in Various Bases
a. Let and 
be two bases in vector space . Every vector 
has a unique representation in basis .
The transpose matrix of system
is called a transformation matrix from basis e to basis f.
b. If P is the matrix of transformation from basis to basis 
. then, for every vector there holds 
, when 
and 
are coordinate vectors of , in bases and respectively.
c. The transformation matrix from basis to basis is invertible, and 
.