Answer to Question #108667 in Physics for MS

Question

Let V=R^3 Determine whether W is a subspace of V where:
W={(a,b,c):a^2+b^2+c^2≤1}

Answer 1

W is a subspace of linear space V, if it is closed under addition and multiplication by a scalar in V. In our case, W is a unit ball at the center of the origin, and both condition are not met, so W is not a subspace of V.

Addition:

For example, let us take two points in W: $\bold x = (0,0,1)$ and $\bold y = (1,0,0)$. Adding them, we obtain $\bold z = \bold x + \bold y = (1,0,1)$. The sum of squared components of this vector is $2$, hence it is not in W.

Multiplication by a scalar:

If we multiply $\bold x = (0,0,1) \in W$ by a number, say $5$, we get $\bold x' = (0,0,5)$, and again, the sum of squared components of this vector is 25, meaning that $\bold x'$ is not in W.