The problem asks to determine if the mapping $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $F(x_1, x_2) = (x_2 + 7, 5x_1)$, is a linear transformation. To determine if $F$ is linear, we need to check if it satisfies the properties of linearity:

1. Additivity: $F(\mathbf{u} + \mathbf{v}) = F(\mathbf{u}) + F(\mathbf{v})$
2. Homogeneity: $F(c \cdot \mathbf{u}) = c \cdot F(\mathbf{u})$

for all vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^2$ and scalars $c \in \mathbb{R}$.

Step-by-Step Solution

1. Check Additivity:

   Let $\mathbf{u} = (x_1, x_2)$ and $\mathbf{v} = (y_1, y_2)$. Then, $\mathbf{u} + \mathbf{v} = (x_1 + y_1, x_2 + y_2).$ Now apply $F$ to $\mathbf{u} + \mathbf{v}$: $F(\mathbf{u} + \mathbf{v}) = F(x_1 + y_1, x_2 + y_2) = ((x_2 + y_2) + 7, 5(x_1 + y_1)).$

   Next, calculate $F(\mathbf{u}) + F(\mathbf{v})$: $F(\mathbf{u}) = (x_2 + 7, 5x_1) \quad \text{and} \quad F(\mathbf{v}) = (y_2 + 7, 5y_1),$ so $F(\mathbf{u}) + F(\mathbf{v}) = (x_2 + 7 + y_2 + 7, 5x_1 + 5y_1) = (x_2 + y_2 + 14, 5x_1 + 5y_1).$

   Since $F(\mathbf{u} + \mathbf{v}) = (x_2 + y_2 + 7, 5x_1 + 5y_1) \neq (x_2 + y_2 + 14, 5x_1 + 5y_1)$, the function does not satisfy additivity.

2. Conclusion:

   Since $F$ fails the additivity property, it is not a linear transformation.

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

Here are some related questions to explore further:

1. What makes a transformation linear in general terms?
2. How would the function $F(x_1, x_2) = (x_2, 5x_1)$ behave under linearity testing?
3. Can a non-linear transformation have a linear part, and if so, what does that mean?
4. How does translation (such as adding a constant) affect linearity?
5. How can we represent linear transformations in matrix form?

Tip: Always check for the additivity and homogeneity properties individually when verifying linearity.