To determine if the transformation $T: \mathbb{E}^2 \rightarrow \mathbb{E}^2$ defined by $T[(x_1, x_2)] = (x_1 + x_2, x_1 - x_2 + 1)$ is a linear transformation, we need to verify if it satisfies the two conditions of linearity:

1. Additivity: $T(\mathbf{v}_1 + \mathbf{v}_2) = T(\mathbf{v}_1) + T(\mathbf{v}_2)$ for any vectors $\mathbf{v}_1$ and $\mathbf{v}_2$.
2. Homogeneity (Scalar Multiplication): $T(c\mathbf{v}) = cT(\mathbf{v})$ for any vector $\mathbf{v}$ and scalar $c$.

Let $\mathbf{v}_1 = (x_1, x_2)$ and $\mathbf{v}_2 = (y_1, y_2)$. Then:

1. Check Additivity:

   Consider $T(\mathbf{v}_1 + \mathbf{v}_2)$. We have: $\mathbf{v}_1 + \mathbf{v}_2 = (x_1 + y_1, x_2 + y_2)$ So, $T(\mathbf{v}_1 + \mathbf{v}_2) = \left( (x_1 + y_1) + (x_2 + y_2), (x_1 + y_1) - (x_2 + y_2) + 1 \right)$ Simplifying: $T(\mathbf{v}_1 + \mathbf{v}_2) = \left( x_1 + x_2 + y_1 + y_2, x_1 - x_2 + y_1 - y_2 + 1 \right)$

   Now, calculate $T(\mathbf{v}_1) + T(\mathbf{v}_2)$. We have: $T(\mathbf{v}_1) = (x_1 + x_2, x_1 - x_2 + 1)$ $T(\mathbf{v}_2) = (y_1 + y_2, y_1 - y_2 + 1)$ Adding these: $T(\mathbf{v}_1) + T(\mathbf{v}_2) = \left( (x_1 + x_2) + (y_1 + y_2), (x_1 - x_2 + 1) + (y_1 - y_2 + 1) \right)$ Simplifying: $T(\mathbf{v}_1) + T(\mathbf{v}_2) = \left( x_1 + x_2 + y_1 + y_2, x_1 - x_2 + y_1 - y_2 + 2 \right)$

   Since $T(\mathbf{v}_1 + \mathbf{v}_2) \neq T(\mathbf{v}_1) + T(\mathbf{v}_2)$, the additivity condition fails. Therefore, $T$ is not a linear transformation.

2. Since the additivity condition fails, we do not need to check homogeneity. $T$ is not linear.

Would you like more details, or do you have any questions?

Here are some related questions:

1. What are examples of linear transformations in 2D space?
2. How do you check if a matrix represents a linear transformation?
3. Can a transformation be partially linear (e.g., linear in one variable)?
4. What is the geometric interpretation of a linear transformation?
5. How do linear transformations relate to vector spaces?

Tip: A linear transformation must preserve both vector addition and scalar multiplication; failing either condition makes the transformation non-linear.