Answer to Question #219806 in Linear Algebra for wisdom

Question

1. Let X = {v1, v2, . . . , vn} be a subset of a vector space V over F. Let

A(X) := {α1v1 + · · · + αnvn | α1 + α2 + · · · + αn = 1}.

Prove that A(X) is a subspace of V if and only if vi = 0V for some i ∈ {1, 2, . . . , n}.

Answer 1

A(x) is the subspace of V only when:

a) A(x) contains zero vector vector vector (matrix)

b) A(x) is closed under matrix addition an matrix multiplication 

Second condition is already fulfilled so A(x) in the subspace of V only when

$vi = 0V \space \space for \space \space some \space \space i ∈ \{1, 2, 3, . . . , n\}$