Abstract Vector Spaces
In this article I want to help you make the leap from and understanding of vectors as perhaps lists of numbers, or arrow, or locations in an n-dimensional plane, to a more abstract understanding of vectors as mathematical objects that can be added together and multiplied by scalars in such a way that they obeys certain axioms, governing their behavior. To understand the full structure of a vector space its necessary to have some limited knowledge of group and field theory. I will not assume any of this knowledge however, and I will explain or link explanations to any concepts that are needed.
Formally, a vector space V over a field F is a set along with two laws of composition, or two operations, known as vector addition and scalar multiplication such that the following conditions hold:
1). The vector space V with addition is an abelian group.
2). For any λ ∈ F, and v ∈ V, the scalar product λv ∈ V and must satisfy distributivity, associativity, and there must be an identity element. ie for λ, k ∈ F, and w, v ∈ V
- λ(v + w) = λv + λw
- (λ + k)w = λw + λk
- λ(kw) = (λk)w
- 1w = w
This defintion may seem like a lot and you might not understand it all but what I want you to take from it is that a vector space is closed under vector addition (the sum of an two vectors must also be a vector in that same space) and scalar multiplication (are vector scaled by some scalar c ∈ F also remains in the space) Note: A vector space must always be define over some field in order for the scalar product to be well-defined.
The beauty of this definition is that it is abstract enough that many very differently “looking” mathematical object can form vector spaces, and thus be treated in the same way. I think the best way to understand this is through examples. I will bring up a few examples of interesting vector spaces, and show some of their properties, but for brevity’s sake I will leave their rigorous verification as vector spaces out in favor of handy-wavy appeals to intuition!
1. ℝⁿ
Let’s begin with the most familiar vector space: V = Rⁿ over some field F which we can take to be R. You can think of Rⁿ as the set of length n tuples, or as a vector in n dimensional space. You can add these vectors together
a + b = (a1, a2, …, an) + (b1 , b2, …, b2) = (a1 + b1, a2 + b2, …, an + bn)
A useful exercise might be showing that lines/planes through origin are subspaces, or at least reasoning why these subsets must contain the origin.
2. The Field of Real Real Numbers
One interesting, but somewhat trivial, example of a vector space is the real numbers over itself. Vector addition is the familiar operation we learn in grade school. And scalar multiplication is just the product of a real number in the vector space, and a scalar (another real number). So it’s just normal scalar multiplication. It clear that the space is closed under both operations, that there exist are identity elements for both operations (what are they?) and in fact that all the conditions for a vector space hold. I would advise trying to show its a vector space. Hint you can get most of the by realizing (R, +) is an abelian group.
3. Space of Real Valued Functions
A more interesting example is the following: let V = F[-a, a] be the set of functions from [-a, a] to the real numbers. I choose an arbitrary interval but this could even be entire real number line. I assert that V over R is a vector space as well where for f,g in V, and λ ∈ R:
- (f + g)(x) = f(x) + g(x)
- (cf)(x) = c(f(x))
Now vectors are real values functions and vector addition is the sum as define above is the sum of values each function maps to for a particular x in the domain. Again it is clear that this space is closed under addition, since the sum of two real valued function on an interval is also a real valued function, and closed under scalar multiplication as well.
Questions:
What is the identity element for each operation?
What is the inverse element for each as well? Hint be careful that the additive inverse/identity are in the vector space while the multiplicative inverse/identity are in the field.
4. Space of Polynomials with Degree ≤ m
Another interesting vector is the polynomial vector space P_m(F) which is the set of all polynomials degree m or less (with coefficient in F). In this space vectors are of the form: p: F→F such that a₀, a₁…a_m ∈ F and
Vector addition on this space is traditional sum of polynomials and scalar multiplication works as you would expect (ie scales up the polynomial coefficients). The sum of two polynomial of degree of no more than m, will never yield a polynomial of degree greater than m. So it is closed under addition. Closure under scalar multiplication is clear. A useful exercise to test your understanding is showing that it closed under additive inverse and identifying the additive identity.
For a polynomial of degree n, this space is n+1 dimensional because of the constant term and is clearly isomorphic to Rⁿ⁺¹ if you take each of the coefficients and arrange them in a vector like so.
Space of Linear Transformations T : V → W
An example of a slightly more abstract vector space is L(V, W) over F, which is the set of all linear maps of the form T : V → W. Vector addition in this space the sum of linear maps such that
holds and scalar multiplication is the product of a scalar and a linear map T such that
holds. You can see that addition of linear maps is defined much the same as the addition of functions! It is clear that adding and scaling linear functions will only create linear functions, and verifying the rest of the properties is a similar exercise to that of my third example for the general real valued function.
