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.