Question

Given a set of vectors S = {v₁ ,…,vn} consider the coordinate linear map

T:Rn → V given by T(x₁ ,…,xn) = x₁ v₁ +x₂ v₂ +⋯+xn vn
​ 1. What statement about T corresponds to the statement V=Span{v₁ ,…,vn} ?

Answer 1

The statement V=Span{v₁ ,…,vn} means that the set of vectors S = {v₁ ,…,vn} spans the vector space V. In other words, every vector in V can be written as a linear combination of the vectors in S.

Step-by-step explanation:

The statement V=Span{v₁ ,…,vn} means that the set of vectors S = {v₁ ,…,vn} spans the vector space V. This means that every vector in V can be written as a linear combination of the vectors in S.

In the given coordinate linear map T:Rn → V, the map T takes a vector (x₁ ,…,xn) in Rn and represents it as a linear combination of the vectors in S. So, if V=Span{v₁ ,…,vn}, it means that every vector in V can be obtained by applying the map T on some vector in Rn.

In other words, if V=Span{v₁ ,…,vn}, it means that the range of the map T is equal to the vector space V.