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 ：\(\textbf{Part (a)}\)

Step 2 ：\(\begin{vmatrix} 1 & -1 & 0 \ 2 & 0 & 1 \ 3 & 1 & 2 \ \end{vmatrix}\)

Step 3 ：\(= 1 \begin{vmatrix} 0 & 1 \ 1 & 2 \ \end{vmatrix} - (-1) \begin{vmatrix} 2 & 1 \ 3 & 2 \ \end{vmatrix} + 0 \begin{vmatrix} 2 & 0 \ 3 & 1 \ \end{vmatrix}\)

Step 4 ：\(= 1(0 - 2) - (-1)(4 - 3)\)

Step 5 ：\(= -2 + 1\)

Step 6 ：\(\boxed{-1}\)

Step 7 ：\(\textbf{Part (b)}\)

Step 8 ：\(\begin{vmatrix} -1 & 3 & 1 \ 1 & 3 & 2 \ 2 & 1 & 2 \ \end{vmatrix}\)

Step 9 ：\(= -1 \begin{vmatrix} 3 & 2 \ 1 & 2 \ \end{vmatrix} - 3 \begin{vmatrix} 1 & 2 \ 2 & 2 \ \end{vmatrix} + 1 \begin{vmatrix} 1 & 3 \ 2 & 1 \ \end{vmatrix}\)

Step 10 ：\(= -1(6 - 2) - 3(2 - 4) + 1(1 - 6)\)

Step 11 ：\(= 4 + 6 - 5\)

Step 12 ：\(\boxed{5}\)

Step 13 ：\(\textbf{Part (c)}\)

Step 14 ：\(\begin{vmatrix} 1 & 0 \ -1 & 1 \ \end{vmatrix}\)

Step 15 ：\(= 1(1) - 0(-1)\)

Step 16 ：\(\boxed{1}\)

Step 17 ：\(\textbf{Part (d)}\)

Step 18 ：\(\begin{vmatrix} 1 & 1 \ 1 & 1 \ \end{vmatrix}\)

Step 19 ：\(= 1(1) - 1(1)\)

Step 20 ：\(\boxed{0}\)