Question

It is like a class discussion and we're supposed to write a discussion about Vector spaces, subspaces and bases

Overview (what to write on the discussion): So we're supposed to discuss and explain about these following points:

1. Vector Spaces, Definitions, Examples, Non Examples
2. Spanning Sets, Linear Independence
3. Basis, Dimension
4. Rank
5. Change of Basis, Coordinates

PUT EXAMPLES AND DEFINITIONS (EX: Showing examples to vector addition (closure under addition, etc) and scalar multiplication (distributive property, etc)

Instructions:

Add a new discussion topic. In your post, please include the following:

1. Details on proofs.
2. Illustrate examples of special interest.
3. Apply your mathematical knowledge of given structures in proving or disproving assertions regarding Vector Spaces.

Answer 1

  

Definition of Spanning Set of a Vector Space: Let S={v1,v2,...vn}S={v1,v2,...vn} be a subset of a vector space VV. The set is called a spanning set of VV if every vector in VV can be written as a linear combination of vectors in SS. In such cases it is said that SS spans VV.

Definition of the span of a set: If S={v1,v2,...vn}S={v1,v2,...vn} is a set of vectors in a vector space VV, then the span of SS is the set of all linear combinations of the vectors in SS, span(S)={k1v1+k2v2+...+knvn|k1,k2,...kn∈R}span(S)={k1v1+k2v2+...+knvn|k1,k2,...kn∈R}. The span of is denoted by span(S)span(S) or span{v1,v2,...vk}span{v1,v2,...vk}. If span(S)=Vspan(S)=V it is said that VV is spanned by {v1,v2,...vn}{v1,v2,...vn}, or that SSspans VV.

What I understand from the definitions:

SS is a subset of the vector space VV and if I can represent all of the vectors that are in the vector space by using just the subset or the smaller part of VV then it can be said that SS spans VV or can reach every vector in VV.

Linear combination has the following form a=k1v1+k2v2+k3v3+...+knvna=k1v1+k2v2+k3v3+...+knvn where kikiare scalars and vivi are the vectors in the subset SSof VV and aa is a particular vector in VV that can be created by a linear combination of vectors in SS. This can be done for infinite number of vectors or all the vectors that are in the vector space VV. We can create a set of all linear combinations of the vectors the can be reached by SS in VV. For instance linear combination aa can be in the set and just like it, many others are a part of this set. We say that SSspans VV if every vector in VV can be reached by the vectors in SS. Furthermore, span(S)span(S) is the set that contains the linear combinations.

Theorem 4.7 Span(S) is a subspace of V: If S={v1,v2,...vn}S={v1,v2,...vn} is a set of vectors in a vector space VV. then span(S)span(S) is a subspace of VV. Moreover, span(S)span(S) is the smallest subspace of VVthat contains SS, in the sense that every other subspace of VV that contains SS must contain span(S)span(S).