A * vector space* is a set of objects

**that follows specific rules of addition and multiplication.**

These objects are called vectors and the rules are:

1) Commutative and associative addition for all elements of the closed set.

2) Associativity and distributivity of scalar multiplication for all elements of the closed set

where and are scalars.

3) Scalar multiplication identity.

4) Additive inverse.

5) Existence of null vector , such that

In a vector space , one vector can be expressed as a ** linear combination** of other vectors in the set, e.g.:

The ** span** of a set of vectors is the set of vectors that can be written as a linear combination of vectors in the set . For example, the span of the set of unit vectors

**and in the space is the set of all vectors (including the null vector) in the space. Alternative, we say that**

**and span .**

If we vary (but not the trivial case where all scalars are zero) such that is equal to ,

the set of vectors is said to be ** linearly dependent** because any vector can be written as a linear combination of the others:

If the only way to satisfy eq1 is when for all , the set of vectors is said to be ** linearly independent**. In this case, we can no longer express any vector as a linear combination of the other vectors (as , resulting in RHS of eq2 being undefined). An example of a set of linearly independent vectors is the set of unit vectors , in the space.

###### Question

Can a set of linearly independent vectors include the zero vector?

###### Answer

No, because if and in eq1, then can be any number. Since is not necessarily 0, it contradicts the definition of linear independence.

A set of linearly independent vectors in an -dimensional vector space forms a set of ** basis vectors**, . A

**basis set is formed by a set of basis vectors of if any vector in the span of can be written as a linear combination of those basis vectors, i.e.**

*complete*