Determine if the described set is a subspace The subset of R

Determine if the described set is a subspace. The subset of R^n (n even) consisting of vectors of the form v = [v1. .. v_n], such that v1 - v2 + v3 - v4 + v5 -. .. - vn = 0. The set is a subspace. The set is not a subspace. If so, give a proof. If not, explain why not.

Solution

It is a subspace

1. Closure under scalar multiplciation

Let v=[v1 v2.... vn]^T be in the set

cv=[cv1 cv2.... cvn]^T

cv1-cv2+....=c(v1-v2+...)=c*0=0

So set is closed under scalar multiplication

2. Closure under addition

Let, v,w be in the set

v=[v1 v2..vn]

w=[w1... wn]

v+w=[(v1+w1) .... (vn+wn)]

(v1+w1)-(v2+2)....=(v1-v2+...)+(w1-w2+...)=0

HEnce closed under addition

Hence the set is a subspace of R^n