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The question asks whether the given transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T\left(\begin{pmatrix} x \\ y \end{pmatrix}\right) = \begin{pmatrix} 1 \\ y \end{pmatrix} \) is linear.

### To determine if a transformation is linear, it must satisfy two properties:
1. **Additivity:** \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \) for any vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^2 \).
2. **Homogeneity:** \( T(c\mathbf{u}) = cT(\mathbf{u}) \) for any vector \( \mathbf{u} \in \mathbb{R}^2 \) and any scalar \( c \).

### Checking Additivity:
Let \( \mathbf{u} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} \).

\[
\mathbf{u} + \mathbf{v} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}
\]

Applying \( T \) to \( \mathbf{u} + \mathbf{v} \):

\[
T(\mathbf{u} + \mathbf{v}) = T\left(\begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}\right) = \begin{pmatrix} 1 \\ y_1 + y_2 \end{pmatrix}
\]

Now applying \( T \) separately to \( \mathbf{u} \) and \( \mathbf{v} \):

\[
T(\mathbf{u}) = \begin{pmatrix} 1 \\ y_1 \end{pmatrix}, \quad T(\mathbf{v}) = \begin{pmatrix} 1 \\ y_2 \end{pmatrix}
\]

\[
T(\mathbf{u}) + T(\mathbf{v}) = \begin{pmatrix} 1 \\ y_1 \end{pmatrix} + \begin{pmatrix} 1 \\ y_2 \end{pmatrix} = \begin{pmatrix} 2 \\ y_1 + y_2 \end{pmatrix}
\]

Clearly, \( T(\mathbf{u} + \mathbf{v}) \neq T(\mathbf{u}) + T(\mathbf{v}) \). Therefore, the transformation is **not** additive.

### Conclusion:
Since the transformation does not satisfy the additivity property, it is **not** linear. The correct answer to the question is **Falso** (False).

Do you want more details or have any questions?

Here are 5 related questions:
1. What is the general form of a linear transformation?
2. How can you determine if a matrix represents a linear transformation?
3. Can a transformation be linear if it only satisfies the homogeneity property?
4. What are some examples of nonlinear transformations?
5. How does the concept of linearity apply to higher-dimensional spaces?

**Tip:** Always verify both the additivity and homogeneity properties when checking for linearity.

Determine si la transformación de V en W dada es lineal. Sea T: R^2 -> R^2 definido por T([x, y]^T) = [1, y]^T.

Learn how to determine whether the transformation T: R^2 -> R^2 defined by T([x, y]^T) = [1, y]^T is linear. Explore the additivity and homogeneity properties crucial for linear transformations, relevant for undergraduate mathematics students.

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