A vector space is a set along with an addition on and a scalar multiplication on such that the following properties hold.
- Commutativity: for all .
- Associativity: and for all and for all .
- Additive Identity: There exists an element such that for .
- Additive Inverse: For every , there exists such that .
- Multiplicative Identity: for all .
- Distributive Properties: and for all and all .
To avoid re-stating this many times, we declare that denotes a vector space over .
Definitions: Addition, Scalar Multiplication
- An addition on a set is a function that assigns an element to each pair of elements .
- A scalar multiplication on a set is a function that assigns an element to each and each .
We can use some geometric language to aid our intuition by saying that elements of a vector space are called vectors or points.
The scalar multiplication in a vector space depends on . Thus when we need to be precise, we will say that is a vector space over instead of saying simply that is a vector space. For example, is a vector space over , and is a vector space over .
- Note that vectors in and scalars in can belong to different sets; the vectors do not need to be elements of the scalar field. For example, in Problem 7 of LADR Exercises 2A, we see the defined as a vector space over . In this case, complex numbers are vectors while real numbers are scalars. The closure of vector addition still applies, as the sum of any two complex numbers is a complex number. The closure of scalar multiplication still applies, as multiplying a complex number by a real number yields another complex number.
Usually the choice of is either clear from the context or irrelevant. Thus we often assume that is lurking in the background without specifically mentioning it. With the usual operations of addition and scalar multiplication, is a vector space over , as you should verify. The example of (see here) motivated our definition of vector space.
The simplest vector space is , which contains only one point.
Elementary Properties
We need to develop some of the elementary properties of vector spaces.
Unique Additive Identity
A vector space has a unique additive identity.
Proof. Suppose and are both additive identities for some vector space . Then
where the first equality holds because is the additive identity, the second equality comes from commutativity, and the third equality holds because is an additive identity. Thus, , showing that only has one additive identity.
Unique Additive Inverse
Every element in a vector space has a has a unique additive inverse.
Proof. Suppose is a vector space. Let . Suppose and are additive inverses of . Then
Thus as desired.
Then we can use the following notation for :
- denotes the additive inverse of ;
- is defined to be .
Multiplication by Scalar 0
When we multiply a vector by the number zero, we have for every .
Proof. For , we have:
Adding the additive inverse of both sides gives:
as desired.
Multiplication by Vector 0
When we multiply a scalar by a vector , we have for every . Here, denotes the additive identity of , not scalar .
Proof. For , we have
Adding the additive inverse of to both sides of the equation above gives , as desired.
Multiplication by -1
Multiplying a vector by gives for every .
Proof. For , we have
This tells us that , when added to , gives . Thus, is the additive inverse of , as desired.