1 Suppose 𝑏, 𝑐 ∈ 𝐑. Define 𝑇∶ 𝐑
3 → 𝐑2 by
𝑇(𝑥, 𝑦, 𝑧) = (2𝑥 − 4𝑦 + 3𝑧 + 𝑏, 6𝑥 + 𝑐𝑥𝑦𝑧).

The linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) is defined by:

\[
T(x, y, z) = \left( 2x - 4y + 3z + b, 6x + cxyz \right)
\]

Where:
- \( T(x, y, z) \) maps a vector \( (x, y, z) \in \mathbb{R}^3 \) to a vector \( (a, b) \in \mathbb{R}^2 \).
- \( b, c \in \mathbb{R} \) are constants.

### Breaking down the transformation:

1. **First component**: 
   \[
   2x - 4y + 3z + b
   \]
   This is a linear combination of \( x \), \( y \), and \( z \) with constant \( b \) added.

2. **Second component**:
   \[
   6x + cxyz
   \]
   This includes a linear term \( 6x \) and a nonlinear term \( cxyz \), where \( c \) is a constant and \( xyz \) represents the product of \( x \), \( y \), and \( z \).

### Is this transformation linear?
To determine if \( T \) is a linear transformation, the following two properties must hold:
- **Additivity**: \( T(u + v) = T(u) + T(v) \) for all \( u, v \in \mathbb{R}^3 \),
- **Homogeneity**: \( T(\alpha u) = \alpha T(u) \) for all \( u \in \mathbb{R}^3 \) and scalars \( \alpha \).

Looking at the second component \( 6x + cxyz \), the term \( cxyz \) is nonlinear due to the product of the variables, which violates the linearity conditions. Hence, **this is not a linear transformation** due to the nonlinear term in the second component.

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Would you like more details on checking linearity or further exploration of this function? Here are five related questions to expand on this:

1. What are the conditions for a function to be a linear transformation?
2. How can you express a linear transformation using a matrix representation?
3. What impact does the nonlinear term \( cxyz \) have on the nature of the transformation?
4. Can we decompose this transformation into linear and nonlinear parts?
5. How would this transformation behave geometrically for specific values of \( b \) and \( c \)?

### Tip:
When dealing with transformations, always verify linearity by checking both additivity and homogeneity conditions.

Suppose 𝑏, 𝑐 ∈ 𝐑. Define 𝑇∶ 𝐑^3 → 𝐑^2 by 𝑇(𝑥, 𝑦, 𝑧) = (2𝑥 − 4𝑦 + 3𝑧 + 𝑏, 6𝑥 + 𝑐𝑥𝑦𝑧).

Explore a transformation from R^3 to R^2 defined by T(x, y, z) = (2x - 4y + 3z + b, 6x + cxyz). This problem analyzes linearity conditions and breaks down both linear and nonlinear components. Suitable for undergraduate mathematics students.

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