Question: $$S\_1=\{(a,a+b,a+b+c)\mid a,b,c\in \mathbb R\}\\
S\_2=\{(1,a+b,a+b+c)\mid a,b,c\in \mathbb R\}\\
S\_3=\left\{(a,b,c)\mid \Bigg|\begin{matrix}a&0&1\\0&a&c\\0&0&b\\\end{matrix}\Bigg|=0\right\}\\
S\_4=\left\{(a,b,c)\mid \Bigg|\begin{matrix}1&0&1\\0&1&0\\a&b&c\\\end{matrix}\Bigg|=0\right\}\\$$
I know that $S\_1$ is a subspace $\mathbb{R}^3$, and $S\_2$ is not a subspace of $\mathbb{R}^3$.

But I'm not sure how to determine if $S\_3$ and $S\_4$ are subspaces of $\mathbb{R}^3$ or not, need some help here thanks!

Answer: 
**hint**
For $S\_3$.

The determinant is $$a^2b$$

$$V\_1=(0,1,1)\in S\_3$$

$$V\_2=(1,0,1)\in S\_3$$

What about $V\_1+V\_2$.
$S\_3$ is not a subspace because the determinant condition implies $a^2b=0$

$S\_4$ is a subspace because the determinant condition implies $a=c$