15. In each part determine whether or not the given set forms a basis for the indicated (sub) space.
(a) $\{(1,2,3),(-1,0,1),(0,1,2)\}$ for ${\mathbb{R}}^3$
(b) $\{(-1,1,2),(3,3,1),(1,2,2)\}$ for ${\mathbb{R}}^3$
(c) $\{(1,-1,0),(0,1,-1)\}$ for the subspace of ${\mathbb{R}}^3$ consisting of all $(x,y,z)$ such that $x+y+z=0$.
(d) $\{(1,1,0),(1,1,1)\}$ for the subspace of ${\mathbb{R}}^3$ consisting of all $(x,y,z)$ such that $y=x+z$.

Step 1 ：$\mathbf{\text{Part (a)}}$

Step 2 ：$\left|\begin{array}{ccccccc}1& -1& 02& 0& 13& 1& 2\end{array}\right|$

Step 3 ：$=1\left|\begin{array}{ccc}0& 11& 2\end{array}\right|-(-1)\left|\begin{array}{ccc}2& 13& 2\end{array}\right|+0\left|\begin{array}{ccc}2& 03& 1\end{array}\right|$

Step 4 ：$=1(0-2)-(-1)(4-3)$

Step 5 ：$=-2+1$

Step 6 ：$\boxed{{\displaystyle -1}}$

Step 7 ：$\mathbf{\text{Part (b)}}$

Step 8 ：$\left|\begin{array}{ccccccc}-1& 3& 11& 3& 22& 1& 2\end{array}\right|$

Step 9 ：$=-1\left|\begin{array}{ccc}3& 21& 2\end{array}\right|-3\left|\begin{array}{ccc}1& 22& 2\end{array}\right|+1\left|\begin{array}{ccc}1& 32& 1\end{array}\right|$

Step 10 ：$=-1(6-2)-3(2-4)+1(1-6)$

Step 11 ：$=4+6-5$

Step 12 ：$\boxed{{\displaystyle 5}}$

Step 13 ：$\mathbf{\text{Part (c)}}$

Step 14 ：$\left|\begin{array}{ccc}1& 0-1& 1\end{array}\right|$

Step 15 ：$=1(1)-0(-1)$

Step 16 ：$\boxed{{\displaystyle 1}}$

Step 17 ：$\mathbf{\text{Part (d)}}$

Step 18 ：$\left|\begin{array}{ccc}1& 11& 1\end{array}\right|$

Step 19 ：$=1(1)-1(1)$

Step 20 ：$\boxed{{\displaystyle 0}}$