The question provides three transformations and asks for the transformed coordinates of points $A(2, 5)$ and $B(-3, 4)$ under these transformations:

1. Transformations:

   • $T_1 = \begin{bmatrix} 4 \\ -1 \end{bmatrix}$
   • $T_2 = \begin{bmatrix} -7 \\ 5 \end{bmatrix}$
   • $T_3 = \begin{bmatrix} -2 \\ 6 \end{bmatrix}$

2. Steps to Solve:

   • For each point, add the transformation vector to the original coordinates to get the new coordinates after each transformation.

Let's calculate the results:

(a) Point $A(2, 5)$

1. Transformation $T_1$: $A' = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \end{bmatrix}$

2. Transformation $T_2$: $A'' = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} -7 \\ 5 \end{bmatrix} = \begin{bmatrix} -5 \\ 10 \end{bmatrix}$

3. Transformation $T_3$: $A''' = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} 0 \\ 11 \end{bmatrix}$

(b) Point $B(-3, 4)$

1. Transformation $T_1$: $B' = \begin{bmatrix} -3 \\ 4 \end{bmatrix} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$

2. Transformation $T_2$: $B'' = \begin{bmatrix} -3 \\ 4 \end{bmatrix} + \begin{bmatrix} -7 \\ 5 \end{bmatrix} = \begin{bmatrix} -10 \\ 9 \end{bmatrix}$

3. Transformation $T_3$: $B''' = \begin{bmatrix} -3 \\ 4 \end{bmatrix} + \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} -5 \\ 10 \end{bmatrix}$

Summary of Results

• For $A(2, 5)$:

  • After $T_1$: $(6, 4)$
  • After $T_2$: $(-5, 10)$
  • After $T_3$: $(0, 11)$

• For $B(-3, 4)$:

  • After $T_1$: $(1, 3)$
  • After $T_2$: $(-10, 9)$
  • After $T_3$: $(-5, 10)$

Would you like additional details on how each transformation affects the points?

Here are 5 follow-up questions:

1. How would the transformations change if the points were located in a three-dimensional space?
2. What is the effect of combining multiple transformations on a single point?
3. How can matrix multiplication be used to represent these transformations?
4. How does applying transformations affect the geometric shape of a figure in coordinate space?
5. What would be the inverse transformations to return the points to their original positions?

Tip: In transformations, always check whether they are translations, rotations, or scalings, as this influences the result significantly.