1. **State the problem:** We have a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$ defined by $$T(x,y,z) = (x + y - z, x - y - z).$$ We want to find a basis for the kernel of $T$, i.e., all vectors $(x,y,z)$ such that $T(x,y,z) = (0,0)$. 2. **Set the system of equations:** Since $T(x,y,z) = (0,0)$, we have $$\begin{cases} x + y - z = 0 \\ x - y - z = 0 \end{cases}$$ 3. **Solve the system:** Add the two equations: $$ (x + y - z) + (x - y - z) = 0 + 0 \implies 2x - 2z = 0 \implies x = z. $$ Substitute $x = z$ into the first equation: $$ z + y - z = y = 0. $$ Thus, $y=0$ and $x=z$. 4. **Express the kernel vectors:** The kernel consists of vectors $(x,y,z)$ with $y=0$ and $x = z$. Let $z = t$, a parameter. Then $$ (x,y,z) = (t,0,t) = t(1,0,1), \quad t \in \mathbb{R}. $$ 5. **Identify the basis:** A basis for the kernel is $\\{(1,0,1)\\}$. **Final answer:** The basis for the kernel of $T$ is option b. $\{(1,0,1)\}$.