The function \( a(x, y, z) = (ax, 0, az) \) defines a vector transformation that operates on a 3-dimensional vector \((x, y, z)\). Here's a breakdown of what it means:

- The function \(a(x, y, z)\) is multiplying the first and third components of the vector by the scalar \(a\), leaving the second component as zero.
- Specifically, if you input a vector \((x, y, z)\), the output vector is \((ax, 0, az)\).

### Key Points:
- **x-component**: The x-coordinate is scaled by \(a\).
- **y-component**: The y-coordinate is always set to 0.
- **z-component**: The z-coordinate is scaled by \(a\).

#### Example:
For \( a = 2 \) and input vector \((3, 4, 5)\):
\[
a(3, 4, 5) = (2 \cdot 3, 0, 2 \cdot 5) = (6, 0, 10)
\]

### Questions for further exploration:
1. How would this transformation affect a 3D object geometrically?
2. What happens if \( a = 0 \)? 
3. Can you find the inverse of this function when \(a \neq 0\)?
4. What is the effect of varying \(a\) continuously on the transformation?
5. How would this transformation affect vectors lying along the x-axis and z-axis?

**Tip**: For vector transformations, understanding how each component changes helps in visualizing the overall geometric effect.

Explore the vector transformation a(x, y, z) = (ax, 0, az) in linear algebra. Understand how scalar multiplication affects each component of the vector. Ideal for students at the undergraduate level.

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