What is the kernel of the linear transformation $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ defined by $T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}-x_{2}, x_{2}+x_{3}\right)$ ?
\[
\operatorname{ker}(T)=\operatorname{span}\{(
\]

Step 1 ：The kernel of a linear transformation T, denoted as ker(T), is the set of all vectors in the domain of T that T maps to the zero vector.

Step 2 ：In this case, we are looking for all vectors (x1, x2, x3) in R^3 such that T(x1, x2, x3) = (0, 0).

Step 3 ：This means we need to solve the system of equations: x1 - x2 = 0 and x2 + x3 = 0.

Step 4 ：The solution to the system of equations is x1 = -x3 and x2 = -x3.

Step 5 ：This means that any vector (x1, x2, x3) in the kernel of T must be of the form (-x3, -x3, x3) for some real number x3.

Step 6 ：We can write this as x3*(-1, -1, 1).

Step 7 ：Therefore, the kernel of T is the set of all scalar multiples of the vector (-1, -1, 1), which is the span of the set {(-1, -1, 1)}.

Step 8 ：\(\boxed{\operatorname{ker}(T)=\operatorname{span}\{(-1, -1, 1)\}}\)