Vector Spaces Help: Definition & Meaning

[christian0710]

Hi i have two questions regarding this definition: "A vector space is a set that is closed under finite vector addition and scalar multiplication"

First of all, is it correctly that a vector space simply is a set of rules that are assigned to a set of vector, the rules are addition and multiplication, and if we apply these rules the set of vecturs are in the vectorspace V?

What does it mean that a set is closed under vector addition and scalar multiplication?

 

[tiny-tim]

hi christian0710!

First of all, is it correctly that a vector space simply is a set of rules that are assigned to a set of vector, the rules are addition and multiplication, and if we apply these rules the set of vecturs are in the vectorspace V?

i don't understand

What does it mean that a set is closed under vector addition and scalar multiplication?

it means the obvious … if you add two vectors (or multiply a vector by a scalar), the result is still a vector in the space

 

[sponsoredwalk]

Start with an arbitrary set of elements, call it V. Then define a special kind of function known as a binary operation on V which we call addition,
⊕ : V × V → V | (v,w) ↦ ⊕(v,w) = v ⊕ w.
This map must satisfy certain obvious & intuitive axioms. Second define a second function known as scalar multiplication,
⊗ : F × V → V | (λ,w) ↦ ⊗(λ,w) = λ ⊗ w,
where F is defined to be a field. This map must also satisfy certain axioms. Thus we have constructed the structure (V,⊕,⊗) known as a vector space. Only within a structure like this can we use the word vector meaningfully, i.e. a vector is an element of the set V in (V,⊕,⊗) that we can manipulate using the maps ⊕ & ⊗ (so the use of the word vector is an informal way of saying that the thing, say v, we're dealing with behaves in a certain manner, i.e. it behaves in the way our axioms allow). The way you've said it is a bit tautological, you can't really speak of a set of vectors until after you've constructed a vector space...

Now as for closure, a function f : S → T is closed if f(x) ∈ T is always true. So for example the function + : {1,2} × {1,2} → {1,2} (which you can think of as a restriction of the addition function on the integers to {1,2} into {1,2}) is not closed on {1,2} because 2 + 2 = 4 ∉ {1,2}. So in the case of vector spaces you'd say that S is closed under vector addition and scalar multiplication if you can construct (S,⊕,⊗), where ⊕ & ⊗ are defined on S satisfying the vector space axioms & the operations are closed on S, basically ⊕(v,w) ∈ S & ⊗(λ,w) ∈ S always holds. This is useful because when you have a vector space (V,⊕,⊗) & you take some arbitrary subset of V, say W, you want to know whether W forms part of a vector space structure on it's own, i.e. (W,⊕,⊗) with ⊕ & ⊗ defined as they were on V (formally you take restrictions of these maps to the set W & want to know whether closure holds, i.e. (W,⊕|ᵂ,⊗|ᵂ), but you don't need to worry about this kind of formality too much).

 

[Fredrik]

Start with an arbitrary set of elements, call it V. Then define a special kind of function known as a binary operation on V which we call addition,
⊕ : V × V → V | (v,w) ↦ ⊕(v,w) = v ⊕ w.
This map must satisfy certain obvious & intuitive axioms. Second define a second function known as scalar multiplication,
⊗ : F × V → V | (λ,w) ↦ ⊗(λ,w) = λ ⊗ w,
where F is defined to be a field. This map must also satisfy certain axioms. Thus we have constructed the structure (V,⊕,⊗) known as a vector space. Only within a structure like this can we use the word vector meaningfully, i.e. a vector is an element of the set V in (V,⊕,⊗) that we can manipulate using the maps ⊕ & ⊗ (so the use of the word vector is an informal way of saying that the thing, say v, we're dealing with behaves in a certain manner, i.e. it behaves in the way our axioms allow). The way you've said it is a bit tautological, you can't really speak of a set of vectors until after you've constructed a vector space...

I agree with all of this. A vector space is a triple (V,⊕,⊗) that satisfies a number of conditions. The set V is called the underlying set of the vector space (V,⊕,⊗). A vector is a member of the underlying set of a vector space.

Now as for closure, a function f : S → T is closed if f(x) ∈ T is always true.

The notation f : S → T means that T is the codomain of f. If T is the codomain of f, then it's always true that f(x)∈T for all x in S. The range f(X)={f(x)|x∈S} is always a subset of the codomain.

So for example the function + : {1,2} × {1,2} → {1,2} (which you can think of as a restriction of the addition function on the integers to {1,2} into {1,2}) is not closed on {1,2} because 2 + 2 = 4 ∉ {1,2}.

The restriction of the addition operation on the integers to {1,2} is a function from {1,2} into the set of integers. Restriction only changes the domain, not the codomain.

What you should be saying here is that the set {1,2} isn't closed under addition, because 1+2=3 isn't in {1,2}.

 

[Fredrik]

First of all, is it correctly that a vector space simply is a set of rules that are assigned to a set of vector, the rules are addition and multiplication, and if we apply these rules the set of vecturs are in the vectorspace V?

I wouldn't say that a vector space is a "set of rules".

Suppose that V is a set, $\mathbb F$ is a field (in pretty much all the interesting examples, $\mathbb F$ is either ℝ or ℂ), A is a map from V×V into V, and S is a map from $\mathbb F$×V into V. The triple (V,A,S) is said to be a vector space if the eight conditions listed here are satisfied. The conditions are written out using the notations A(x,y)=x+y and S(a,x)=ax.

If (V,A,S) is a vector space,

• the map A is called addition, and we use the notation x+y instead of A(x,y).
• the map S is called scalar multiplication, and we use the notation ax instead of S(a,x).
• the set V is called the underlying set of the vector space (V,A,S).
• the members of V are called vectors.
• the members of $\mathbb F$ are called scalars.

What does it mean that a set is closed under vector addition and scalar multiplication?

Addition is a function $V\times V\to V$. Scalar multiplication is a function $\mathbb F\times V\to V$, where $\mathbb F$ is a field. A set $S\subset V$ is said to be closed under addition if x+y is in S for all x,y in S. A set $S\subset V$ is said to be closed under scalar multiplication if λx is in S for all λ in $\mathbb F$ and all x in S.

 

[christian0710]

Thank you guys! I'm on the way out, but will be studying and reading it this weekend. I appreciate your help in advance!