The problem presents three mathematical statements and asks which of them are always true. Here's an analysis of each statement:

Statement I:

The set $\{(x, y, z) \mid x + y \leq 0\}$ is a subspace of $\mathbb{R}^3$.
Analysis:
For a set to be a subspace of $\mathbb{R}^3$, it must satisfy the following conditions:

1. It must include the zero vector $(0, 0, 0)$,
2. It must be closed under vector addition,
3. It must be closed under scalar multiplication.

Let us test:

• The zero vector $(0, 0, 0)$ satisfies $x + y = 0 + 0 = 0 \leq 0$, so it is included.
• For any $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in the set, $x_1 + y_1 \leq 0$ and $x_2 + y_2 \leq 0$, but their sum $(x_1 + x_2) + (y_1 + y_2)$ could violate this inequality.
  For example: $(x_1, y_1, z_1) = (-1, 1, 0)$ and $(x_2, y_2, z_2) = (-1, 1, 0)$. While both satisfy $x + y \leq 0$, their sum is $(-2, 2, 0)$, where $-2 + 2 = 0 \not\leq 0$.

Thus, it is not a subspace of $\mathbb{R}^3$.

Statement II:

The transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ given by $T(x, y) = (x + 2, y)$ is a linear transformation.
Analysis:
To check if a function $T(x, y)$ is linear:

1. It must satisfy $T(u + v) = T(u) + T(v)$,
2. It must satisfy $T(cu) = cT(u)$, where $u, v \in \mathbb{R}^2$ and $c \in \mathbb{R}$.

Here,
$T(x, y) = (x + 2, y)$. Test the zero vector:

• $T(0, 0) = (0 + 2, 0) = (2, 0) \neq (0, 0)$.

Since $T(0) \neq 0$, the function is not linear.

Statement III:

The transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$, given by $T(x, y, z) = (0, x + z, y)$, maps the plane $y = 0$ to the $y$-axis.
Analysis:
For any point $(x, 0, z)$ on the plane $y = 0$,
$T(x, 0, z) = (0, x + z, 0)$.

The image under $T$ always lies on the $y$-axis because the second coordinate $(x + z)$ corresponds to the $y$-axis, while the first and third coordinates are always $0$.
Thus, this statement is true.

Answer:

Only Statement III is always true.
The correct choice is C) III.

Do you want further explanation, or do you have any additional questions?
Here are some related questions to explore:

1. What conditions must be satisfied for a set to be a subspace of $\mathbb{R}^n$?
2. What are the properties of linear transformations?
3. How can you determine the image of a plane under a linear transformation?
4. How can the zero vector test verify a subspace or linear transformation?
5. Can you provide another example of a transformation that isn't linear?

Tip: Always check the zero vector and operations (addition/scalar multiplication) to verify linearity and subspaces.