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.