verify if T:E^2$$ \rarr E^2 $$ DEFINed by T[(x1,x2)]=(x1+x2,x1-x2+1) is linear transformation or not

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.

Explore whether the transformation T: E^2 to E^2 defined by T[(x1,x2)] = (x1 + x2, x1 - x2 + 1) is a linear transformation. Understand the criteria for linearity including additivity and homogeneity. Detailed mathematical analysis provided.

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