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.