To determine if the function $F: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $F(x, y, z) = (x, z)$ is linear, we need to verify two key properties of linearity:

1. Additivity: $F(\mathbf{u} + \mathbf{v}) = F(\mathbf{u}) + F(\mathbf{v})$
2. Homogeneity (or scalar multiplication): $F(c \mathbf{u}) = c F(\mathbf{u})$

Step 1: Check Additivity

Let $\mathbf{u} = (x_1, y_1, z_1) \in \mathbb{R}^3$ and $\mathbf{v} = (x_2, y_2, z_2) \in \mathbb{R}^3$. We calculate:

$F(\mathbf{u} + \mathbf{v}) = F((x_1 + x_2, y_1 + y_2, z_1 + z_2)) = (x_1 + x_2, z_1 + z_2)$ Now, compute the sum of $F(\mathbf{u})$ and $F(\mathbf{v})$:

$F(\mathbf{u}) = (x_1, z_1), \quad F(\mathbf{v}) = (x_2, z_2)$ $F(\mathbf{u}) + F(\mathbf{v}) = (x_1, z_1) + (x_2, z_2) = (x_1 + x_2, z_1 + z_2)$ Since $F(\mathbf{u} + \mathbf{v}) = F(\mathbf{u}) + F(\mathbf{v})$, the function satisfies the additivity property.

Step 2: Check Homogeneity

Let $c \in \mathbb{R}$ and $\mathbf{u} = (x, y, z) \in \mathbb{R}^3$. We calculate:

$F(c \mathbf{u}) = F(c(x, y, z)) = F((cx, cy, cz)) = (cx, cz)$ Now, compute $c F(\mathbf{u})$:

$F(\mathbf{u}) = (x, z), \quad c F(\mathbf{u}) = c(x, z) = (cx, cz)$ Since $F(c \mathbf{u}) = c F(\mathbf{u})$, the function satisfies the homogeneity property.

Conclusion:

Since the function $F(x, y, z) = (x, z)$ satisfies both the additivity and homogeneity properties, it is linear.

Would you like more details or have any questions?

Here are 5 related questions to explore this concept further:

1. What is the geometric interpretation of linear functions between vector spaces?
2. How does the matrix representation of a linear function look in this case?
3. What are the properties of the kernel and image of this function?
4. Can you describe the inverse of a linear function? Does this function have an inverse?
5. How does this function behave when applied to basis vectors?

Tip: Always check both properties (additivity and homogeneity) when determining if a function is linear.