The difference between vector space and groupWhen comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions?1. Why we need to define a scalar product for a vector space? Physical sense or some insight behind it?2. One truly nice thing for vector space is that we represent the element with basis, so what we do with elements in vector space is just with basis,so why we can't do the same thing for group?

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The difference between vector space and groupWhen comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions?1. Why we need to define a scalar product for a vector space? Physical sense or some insight behind it?2. One truly nice thing for vector space is that we represent the element with basis, so what we do with elements in vector space is just with basis,so why we can&#039;t do the same thing for group?

lelestalis80d · Accepted Answer

Step 1As you may know, a vector space is a set V together with operations   +  :  V  ×  V  →  V and   ⋅  :  K  ×  V  →  V that satisfy certain conditions, where K is a field (take   K  =      R   for instance). Turns out that these conditions makes   (  V  ,  +  ) into an abelian group, a fancy term for a commutative group. This means that if you take V and remove the scalar multiplication operator, the elements of V forms a group and commute with each others.Conversely, you can take an abelian group and try to turn it into a vector space by adding scalar multiplication on it. This additional structure comes in handy when you want to reason about lengths and angles of vectors in V. A geometric interpretation of this is that it stretches, or contracts, vectors   v  ∈  V by a constant factor   α  ∈  K. In fact, scalars scale vectors.Step 2Without scalar multiplication, it is not possible think of any way of constructing a basis in a group G. If you think back of the definition of a basis, you will see that it involves a field. The definition of a vector space encapsulates the notion of basis in some sense.

When comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions? 1. Why we need to define a scalar product for a vector space? Physical sense or some insight behind it?

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