Answer to Question #282169 in Linear Algebra for JOY

Question

Determine wheather the following sets are subspaces of R

3

{(a,b,c) : a

2+ b

2+ c

2 ≤1, a,b,c ∈R}

Answer 1

Let us determine wheather the set $W=\{(a,b,c) : a^2+ b^2+ c^2 ≤1, a,b,c ∈\R\}$ is a subspace of $\R^3.$ Taking into account that $1^2+0^2+0^2=1\le1,$ we conclude that $(1,0,0)\in W.$ On the other hand, $(1,0,0)+(1,0,0)=(2,0,0)\notin W$ because of $2^2+0^2+0^2=4\ge1.$ Therefore, $W$ is not a subspace of $\R^3.$