Question

V=R∞
W={x⃗ 1,x⃗ 2,x⃗ 3,...}:x⃗ n+2=x⃗ n+1+x⃗ n;n≥1
is W a subspace of V ?

Answer 1

actually no, I think not closed under addition
let's add two vectors and call the result y\[y=x_5+x_3=x_4+2x_3=3x_3+x_2=4x_2+2x_1\]which is not of the form prescribed, so y is not in W... I think
I hope that's right, but I won't pretend I'm positive
definitely seems closed under scalar multiplication though

Answer 2

thank you soo much!!

Answer 3

welcome!
hope that's right :P