Let's analyze the problem and determine whether the transformation $T$, defined by:

$T(x_1, x_2) = \begin{pmatrix} 2x_1 - 3x_2 \\ x_1 + 4 \\ 4x_2 \end{pmatrix}$ is linear or not.

Conditions for Linearity

A transformation $T$ is linear if it satisfies the following two properties:

1. Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. Homogeneity (Scaling): $T(c\mathbf{u}) = cT(\mathbf{u})$ for any scalar $c$.

Step 1: Test Zero Vector

First, we must check what $T(\mathbf{0})$ equals. If $T$ is linear, then: $T(0, 0) = 0$

Substitute $x_1 = 0$ and $x_2 = 0$ into the transformation: $T(0, 0) = \begin{pmatrix} 2(0) - 3(0) \\ 0 + 4 \\ 4(0) \end{pmatrix} = \begin{pmatrix} 0 \\ 4 \\ 0 \end{pmatrix}$

Since $T(0, 0) \neq \mathbf{0}$, this shows that $T$ is not linear, because a linear transformation must map the zero vector to the zero vector.

Conclusion

The transformation $T$ is not linear because it does not satisfy the condition $T(0) = 0$.

Would you like more details or have any questions on this? Here are some related questions to expand on this topic:

1. What are some examples of linear transformations?
2. Can a transformation be non-linear if it satisfies additivity but not homogeneity?
3. How does the concept of linearity apply to higher-dimensional transformations?
4. What is the importance of the zero vector in determining linearity?
5. Can a non-linear transformation be approximated by a linear one under certain conditions?

Tip: Always check the behavior of the zero vector when verifying if a transformation is linear!