- Our vectors, which we denote as usual by greek letters , are the positive real numbers, .
- Our field of scalars (denoted by roman letters
can be either the reals or the rationals . We'll write but
we get a vector space with as well.
- To ``add'' two vectors, we multiply the corresponding real numbers. That is, the vector sum of and is the product .
- The scalar product of a scalar and the vector , we compute the power .
Let's check that this satisfies the necessary properties:
- additive closure
- If and are positive real numbers, so is their product .
- associativity of addition
- Multiplication of real numbers is associative, so no problem.
- commutivity of addition
- Multiplication of real numbers is commutative.
- additive identity
- The positive real number 1 acts as the identity element for vector addition, since .
- additive inverses
- For any vector
, there is another
so when the two vectors are ``added'', the result
is the identity element above. In this case, the inverse of is
- closure of scalar multiplication
- For any scalar and any vector , the scalar multiple is still in .
- neutrality of 1
- When we compute the scalar multiple of the
multiplicative identity in our field with any vector
we should get the original vector. That works fine:
- vector distributive law
- Multiplying a scalar times the sum of two
vectors and :
- scalar distributive law
- The sum of two scalars times a vector
So we see that is a vector space over , with an appropriate interpretation of vector addition and scalar multiplication.
Note also that in this case, a ``linear combination'' works out to be very like factoring. For example, we can express the vector as a linear combination of the vectors and by
Note that there may be more than one way to express the same vector as a
linear combination of two others. For example, if our underlying field is
, then there are scalars in equal to and , and
is also a scalar. But then if we have
we also have
More concretely, since
we also have
So we can express as a linear combination of and in many different ways. What is the dimension of as a vector space over ? Can you give a proof?
The situation is more complicated if we consider as a vector space over , since will be rational for some and , and not for others. What do you think the dimension of this vector space is?