2025-01-12

946: Product Vectors Space

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definition of product vectors space

Topics


About: vectors space

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of product vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description 1


Here is the rules of Structured Description.

Entities:
J: ∈{ the possibly uncountable index sets }
F: ∈{ the fields }
{Vj|j∈J}: ⊆{ the F vectors spaces }
∗×j∈JVj: = the product set , ∈{ the F vectors spaces }, with the operations specified below
//

Conditions:
∀r∈F,∀f∈×j∈JVj(∀j∈J((rf)(j)=r(f(j))))
∧
∀f,f′∈×j∈JVj(∀j∈J((f+f′)(j)=f(j)+f′(j)))
//

2: Structured Description 2


Here is the rules of Structured Description.

Entities:
F: ∈{ the fields }
{V1,...,Vk}: ⊆{ the F vectors spaces }
∗V1×...×Vk: = the product set , ∈{ the F vectors spaces }, with the operations specified below
//

Conditions:
∀r∈F,∀v=(v1,...,vk)∈V1×...×Vk(rv=(rv1,...,rvk))
∧
∀v=(v1,...,vk),v′=(v1′,...,vk′)∈V1×...×Vk(v+v′=(v1+v1′,...,vk+vk′))
//

3: Note


Let us see that ×j∈JVj is indeed an F vectors space.

1) for any elements, f,f′∈×j∈JVj, f+f′∈×j∈JVj (closed-ness under addition): for each j∈J, (f+f′)(j)=f(j)+f′(j)∈Vj.

2) for any elements, f,f′∈×j∈JVj, f+f′=f′+f (commutativity of addition): for each j∈J, (f+f′)(j)=f(j)+f′(j)=f′(j)+f(j)=(f′+f)(j).

3) for any elements, f,f′,f″∈×j∈JVj, (f+f′)+f″=f+(f′+f″) (associativity of additions): for each j∈J, ((f+f′)+f″)(j)=(f+f′)(j)+f″(j)=f(j)+f′(j)+f″(j)=f(j)+(f′+f″)(j)=(f+(f′+f″))(j).

4) there is a 0 element, 0∈×j∈JVj, such that for any f∈×j∈JVj, f+0=f (existence of 0 vector): f0∈×j∈JVj such that for each j∈J, f0(j)=0, is 0, because (f+f0)(j)=f(j)+f0(j)=f(j).

5) for any element, f∈×j∈JVj, there is an inverse element, f′∈×j∈JVj, such that f′+f=0 (existence of inverse vector): f′∈×j∈JVj such that for each j∈J, f′(j)=−f(j), is such a one, because (f′+f)(j)=f′(j)+f(j)=−f(j)+f(j)=0.

6) for any element, f∈×j∈JVj, and any scalar, r∈F, r.f∈×j∈JVj (closed-ness under scalar multiplication): (r.f)(j)=rf(j)∈Vj.

7) for any element, f∈×j∈JVj, and any scalars, r1,r2∈F, (r1+r2).f=r1.f+r2.f (scalar multiplication distributability for scalars addition): for each j∈J, ((r1+r2).f)(j)=(r1+r2)f(j)=r1f(j)+r2f(j)=(r1f)(j)+(r2f)(j)=(r1.f+r2.f)(j).

8) for any elements, f,f′∈×j∈JVj, and any scalar, r∈F, r.(f+f′)=r.f+r.f′ (scalar multiplication distributability for vectors addition): for each j∈J, (r.(f+f′))(j)=r(f+f′)(j)=r(f(j)+f′(j))=rf(j)+rf′(j)=(rf)(j)+(rf′)(j)=(r.f+r.f′)(j).

9) for any element, f∈×j∈JVj, and any scalars, r1,r2∈F, (r1r2).f=r1.(r2.f) (associativity of scalar multiplications): for each j∈J, ((r1r2).f)(j)=(r1r2)f(j)=r1(r2f(j))=r1((r2f)(j))=(r1.(r2.f))(j).

10) for any element, f∈×j∈JVj, 1.f=f (identity of 1 multiplication): for each j∈J, (1.f)(j)=1.(f(j))=f(j).

Let us see that V1×...×Vk is indeed an F vectors space.

1) for any elements, v,v′∈V1×...×Vk, v+v′∈V1×...×Vk (closed-ness under addition): for v=(v1,...,vk) and v′=(v1′,...,vk′), v+v′=(v1+v1′,...,vk+vk′)∈V1×...×Vk.

2) for any elements, v,v′∈V1×...×Vk, v+v′=v′+v (commutativity of addition): for v=(v1,...,vk) and v′=(v1′,...,vk′), v+v′=(v1,...,vk)+(v1′,...,vk′)=(v1+v1′,...,vk+vk′)=(v1′+v1,...,vk′+vk)=(v1′,...,vk′)+(v1,...,vk)=v′+v.

3) for any elements, v,v′,v″∈V1×...×Vk, (v+v′)+v″=v+(v′+v″) (associativity of additions): for v=(v1,...,vk), v′=(v1′,...,vk′), and v″=(v1″,...,vk″), (v+v′)+v″=(v1+v1′,...,vk+vk′)+(v1″,...,vk″)=(v1+v1′+v1″,...,vk+vk′+vk″)=(v1,...,vk)+(v1′+v1″,...,vk′+vk″)=v+(v′+v″).

4) there is a 0 element, 0∈V1×...×Vk, such that for any v∈V1×...×Vk, v+0=v (existence of 0 vector): v0=(0,...,0)∈V1×...×Vk is 0, because for v=(v1,...,vk), v0+v=(0+v1,...,0+vk)=(v1,...,vk)=v.

5) for any element, v∈V1×...×Vk, there is an inverse element, v′∈V1×...×Vk, such that v′+v=0 (existence of inverse vector): for v=(v1,...,vk), v′=(−v1,...,−vk)∈V1×...×Vk is a one, because v′+v=(−v1,...,−vk)+(v1,...,vk)=(−v1+v1,...,−vk+vk)=(0,...,0)=0.

6) for any element, v∈V1×...×Vk, and any scalar, r∈F, r.v∈V1×...×Vk (closed-ness under scalar multiplication): for v=(v1,...,vk), r.v=(rv1,...,rvk)∈V1×...×Vk.

7) for any element, v∈V1×...×Vk, and any scalars, r1,r2∈F, (r1+r2).v=r1.v+r2.v (scalar multiplication distributability for scalars addition): for v=(v1,...,vk), (r1+r2).v=((r1+r2)v1,...,(r1+r2)vk)=(r1v1+r2v1,...,r1vk+r2vk)=(r1v1,...,r1vk)+(r2v1,...,r2vk)=r1(v1,...,vk)+r2(v1,...,vk)=r1.v1+r2.v2.

8) for any elements, v,v′∈V1×...×Vk, and any scalar, r∈F, r.(v+v′)=r.v+r.v′ (scalar multiplication distributability for vectors addition): for v=(v1,...,vk) and v′=(v1′,...,vk′), r.(v+v′)=r(v1+v1′,...,vk+vk′)=(r(v1+v1′),...,r(vk+vk′))=(rv1+rv1′,...,rvk+rvk′)=(rv1,...,rvk)+(rv1′,...,rvk′)=r(v1,...,vk)+r(v1′,...,vk′)=r.v+r.v′.

9) for any element, v∈V1×...×Vk, and any scalars, r1,r2∈F, (r1r2).v=r1.(r2.v) (associativity of scalar multiplications): for v=(v1,...,vk), (r1r2).v=((r1r2)v1,...,(r1r2)vk)=(r1(r2v1),...,r1(r2vk))=r1(r2v1,...,r2vk)=r1.(r2.v).

10) for any element, v∈V1×...×Vk, 1.v=v (identity of 1 multiplication): for v=(v1,...,vk), 1.v=(1v1,...,1vk)=(v1,...,vk)=v.

References
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