To determine which of the transformations are linear, we need to use the definition of linearity. A transformation $T: V \to W$ is linear if it satisfies two properties for all vectors $u, v \in V$ and scalars $c \in \mathbb{R}$:

1. Additivity (or superposition): $T(u + v) = T(u) + T(v)$
2. Homogeneity (or scalar multiplication): $T(cu) = cT(u)$

Let's analyze each transformation from the image:

1. $T: \mathbb{R} \to \mathbb{R}: T(x) = \sin(x)$

   • Not linear, because $\sin(x)$ is not a linear function. For example, $\sin(x + y) \neq \sin(x) + \sin(y)$.

2. $T: \mathbb{R} \to \mathbb{R}: T(x) = 2x + 1$

   • Not linear, because of the constant term $+1$. The transformation does not satisfy homogeneity (e.g., $T(2x) = 2(2x + 1)$, not equal to $2T(x)$).

3. $T: \mathbb{R}^2 \to \mathbb{R}^3: T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = x_1 \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} + x_2 \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$

   • Linear, as this is a matrix transformation. It satisfies both additivity and homogeneity because it is a linear combination of the input components.

4. $T: \mathbb{R}^3 \to \mathbb{R}^3: T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - 2x_2 + x_3 \\ 4x_1 + x_2 - 2x_3 \\ -x_1 + 2x_2 - x_3 \end{bmatrix}$

   • Linear, as it is expressed as a linear combination of its inputs without any constant terms.

5. $T: \mathbb{R}^2 \to \mathbb{R}^3: T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + x_1 \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} + x_2 \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix}$

   • Not linear, because of the constant vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$. This violates homogeneity.

6. $T: \mathbb{R}^3 \to \mathbb{R}: T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \cdot \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$

   • Linear, as it is a dot product, which is a linear operation.

Correct answers:

• The third transformation is linear.
• The fourth transformation is linear.
• The last transformation (sixth) is linear.

Do you want more details or have any questions?

Here are 5 questions related to this topic:

1. What is the definition of a linear transformation?
2. How can you represent a linear transformation using matrices?
3. What are the conditions that a transformation must satisfy to be considered linear?
4. Can non-linear transformations be expressed in matrix form?
5. Why do constant terms in a transformation prevent it from being linear?

Tip: Checking if a transformation satisfies both additivity and homogeneity is the easiest way to determine linearity.