In this section, we will explore the concepts of vector spaces and subspaces. These are very important to our understanding of linear algebra. In fact, if we do not have an understanding of vector spaces and subspaces, we do not truly have an understanding of how to solve linear algebra problems.
Vector spaces and subspaces
Spaces
Vector spaces are one of the fundamental settings for linear algebra, and, as the name suggests, they are spaces where all vectors reside. We will denote the vector space with V.
The easiest way to think of dimensions is to count the number of elements in the column vector. Suppose we have 
, then 
. 
is a straight line, 
is all the possible points in the xy-plane, and 
is all the possible points in the xyz-plane—that is, 3-dimensional space, and so on.
The following are some of the rules for vector spaces:
- There exists in V an additive identity element such that
for all 
. - For all
, there exists an additive inverse such that 
. - For all
, there exists a multiplicative identity such that 
. - Vectors are commutative, such that for all
, 
. - Vectors are associative, such that
. - Vectors have distributivity, such that
and 
for all 
and for all 
.
A set of vectors is said to be linearly independent if 
, which implies that 
.
Another important concept for us to know is called span. The span of 
is the set of all linear combinations that can be made using the n vectors. Therefore, 
if the vectors are linearly independent and span V completely; then, the vectors 
are the basis of V.
Therefore, the dimension of V is the number of basis vectors we have, and we denote it dimV.
Subspaces
Subspaces are another very important concept that state that we can have one or many vector spaces inside another vector space. Let's suppose V is a vector space, and we have a subspace 
. Then, S can only be a subspace if it follows the three rules, stated as follows:
and 
, which implies that S is closed under addition
and 
so that 
, which implies that S is closed under scalar multiplication
If 
, then their sum is 
, where the result is also a subspace of V.
The dimension of the sum 
is as follows:
