Let's analyze the problem step by step:

Problem Breakdown

1. The initial triangle $\triangle ABC$ is defined with vertices:

   • $A(1, 0)$
   • $B(1, 2)$
   • $C(3, 1)$.

2. The linear transformation $T_A : \mathbb{R}^2 \to \mathbb{R}^2$ is defined using a matrix: $M = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}.$ This means the transformation applies scaling by a factor of 2 in both $x$- and $y$-directions.

3. The transformed coordinates $T_A((x, y))$ are calculated using matrix multiplication: $T_A((x, y)) = M \cdot \begin{bmatrix} x \\ y \end{bmatrix}.$

4. We need to:

   • Determine the transformed coordinates $A'$, $B'$, and $C'$,
   • Represent the transformed triangle $\triangle A'B'C'$ on the graph.

Solution Steps

Transformation of Vertices

Using the matrix $M = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$, we apply the transformation $T_A((x, y))$ to each vertex:

1. For $A(1, 0)$: $T_A((1, 0)) = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}.$ So, $A'(2, 0)$.

2. For $B(1, 2)$: $T_A((1, 2)) = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}.$ So, $B'(2, 4)$.

3. For $C(3, 1)$: $T_A((3, 1)) = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 6 \\ 2 \end{bmatrix}.$ So, $C'(6, 2)$.

Final Transformed Vertices

The transformed vertices are:

• $A'(2, 0)$,
• $B'(2, 4)$,
• $C'(6, 2)$.

Graphical Representation

The transformed triangle $\triangle A'B'C'$ will be larger than $\triangle ABC$, with its vertices located at the above-calculated points.

Would you like a detailed sketch of the transformed triangle or further explanation about the transformation process?

Expanding the Topic

Here are five related questions you could explore:

1. What happens to the shape of the triangle if a shear transformation matrix is applied instead of scaling?
2. How would the transformation change if the scaling factor were different in $x$- and $y$-directions?
3. What is the determinant of the transformation matrix, and what does it signify geometrically?
4. How do linear transformations affect the area of geometric shapes like triangles?
5. What would the result be if the matrix $M$ included rotation components?

Tip

When working with transformations, always visualize the effect of the matrix on the unit vectors $(1, 0)$ and $(0, 1)$. This helps understand the overall geometric impact of the transformation.