6. Let W= (a): a + b + c = 0). Is W a subspace of R^3
(b)
(c)
the question must be,
$\displaystyle W=\left \{ (a,b,c)\in \mathbb{R}^{3}\mid a+b+c=0 \right \} $ is it a subspace of $\displaystyle \mathbb{R}^{3}$ ?
Now can you answer these questions ?
is it closed under addition ?
is it closed under scalar multiplication ?
abc is not horizontal, it's vertical if it changes anything. And to be honest I have no idea how to do this problem. My teacher had this problem on the homework and we literally did not learn anything yet. I would ask for the answer but it's not fair to ask that on here.
Let $\displaystyle X=(a,b,c)$ and $\displaystyle X'=(a',b',c')$ elements of $\displaystyle W$.We have also $\displaystyle X+X'= (a+a',b+b',c+c') $ element of $\displaystyle W$.Indeed,
$\displaystyle (a+a')+(b+b')+(c+c')=(a+b+c)+(a'+b'+c')=0$
This proves that $\displaystyle W$ is closed under addition,try to prove that is also closed under scalar multiplication.