The strategy is to create a vector space and then introduce to you the facts that must be true for that space to truly be a vector space.
Informal definition of Vector Space
Let V be the space where vectors can be found, and, using your mathematical superpowers, impose two operations, + and *, on V, along with a short list of rules.
Addition, + is an operation for which operating on two vectors from V results in a vector from V. If this is true then V is closed to +.
Multiplication, *, is an operation that takes a scalar and a vector and the result is a vector.
Addition is Commutative and Associative, it has a unique identity element and additive Inverse Elements exists.
Multiplication is associative, Scalar Multiplication distributes over Vector Addition, there is Vector Distributivity over Scalar Addition and it has a unique identity element.
A vector is an element that exists in the above described vector space.
We will be drawing arrows from the origin to points in the vector space. One author mentions that this is a subset of what is possible. For our system, for every point in the vector space there exists a vector whose tail is at the origin and its head is at this point.
Vector addition takes two vectors and places them head to tail to create a third vector.
Appendix A
A fair question ask, after seeing all the specifics piled together to build a vector space–can we define something as just being a Space, that has a simpler definition?
Appendix B
One author grouped properties of a Vector Space into three categories: Addition Properties, Scaling Properties, Distribution Properties.
Addition Properties
- Sum of two vectors is a vector
- {v, w, v+w} elements of vector space
- Vector addition is commutative
- v+w = w+v
- Vector addition is associative
- u+(v+w) = (u+v)+w
- Zero Element (Additive Identity Element) exists
- v+0 = v
- Additive Inverse Element Exists
- v + (-v) = 0
Scaling Properties
- Scaled vector is also a vector
- {v, av} elements of vector space
- Scaling a vector with a first scalar and taking that result and scaling it with a second scalar gives the same result as multiplying those two scalars together to get a product scalar and scaling the vector with that product scalar
- a(bv) = (ab)v
- Existence of Unity Element (Existence of Multiplicative Identity)
- 1v = v
Distributive Properties
- A scaled sum equals the sum of the scaled vectors
- a(v + w) = av + aw
- A vector scaled with the sum of scalars equals the sum vectors scaled with each scalar
- (a + b)v = av + bv
Vectoreverything 