Linear Operators:
Linear operators are functions that operate on vectors and satisfy certain properties. In this question, we have two linear operators, T1 and T2, defined on R^3.

T1 Operator:
The T1 operator is defined as T1(x, y, z) = (x, x + y, x - y - z). Let's break down what this operator does to a given vector (x, y, z).

1. The first component of the output vector is simply x, which means it remains unchanged.
2. The second component of the output vector is x + y, which is the sum of the first two components of the input vector.
3. The third component of the output vector is x - y - z, which is the difference between the first and second components of the input vector, subtracted by the third component.

T2 Operator:
The T2 operator is defined as T2[x, y, z] = |x 2z, y - z, x y z|. Here, the "|" symbol denotes the determinant of a 3x3 matrix.

1. The first row of the matrix is (x, 2z, 0), where the second component is 2z.
2. The second row of the matrix is (y - z, 0, 0), where the first component is y - z and the other components are zero.
3. The third row of the matrix is (x, y, z), which is the same as the input vector.

Now, let's evaluate the determinant of this matrix.

Determinant = (x * 0 * z + 2z * 0 * x + 0 * (y - z) * x) - (0 * 2z * z + x * (y - z) * 0 + (y - z) * 0 * x)
= 0 - 0 + 0
= 0

Summary:
In summary, we have two linear operators, T1 and T2, defined on R^3. The T1 operator transforms a vector (x, y, z) into (x, x + y, x - y - z), while the T2 operator transforms a vector (x, y, z) into a matrix with a determinant of 0.