Let's solve the given problems by finding the inverses of the linear transformations provided.

Problem (a)

The linear transformation $T_1: \mathbb{R}^2 \to \mathbb{R}^2$ is given by: $T_1(x, y) = (2x + 5y, 6x + 16y)$

To find the inverse transformation $T_1^{-1}(x, y)$, we need to express the variables $x$ and $y$ in terms of the output coordinates.

1. Represent the transformation in matrix form: $T_1(x, y) = \begin{bmatrix} 2 & 5 \\ 6 & 16 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

2. Find the inverse of the matrix $A = \begin{bmatrix} 2 & 5 \\ 6 & 16 \end{bmatrix}$.

The inverse of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is: $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

3. Compute the determinant of $A$: $\text{det}(A) = (2)(16) - (5)(6) = 32 - 30 = 2$

4. The inverse matrix $A^{-1}$ is: $A^{-1} = \frac{1}{2} \begin{bmatrix} 16 & -5 \\ -6 & 2 \end{bmatrix} = \begin{bmatrix} 8 & -\frac{5}{2} \\ -3 & 1 \end{bmatrix}$

5. Thus, the inverse transformation $T_1^{-1}(x, y)$ is: $T_1^{-1}(x, y) = \left( 8x - \frac{5}{2}y, -3x + y \right)$

Problem (b)

The linear transformation $T_2: \mathbb{R}^3 \to \mathbb{R}^3$ is given by: $T_2(x, y, z) = (x + 2z, 2x + y, 2y + z)$

To find the inverse transformation $T_2^{-1}(x, y, z)$, we again need to find the inverse of the corresponding matrix.

1. Represent the transformation in matrix form: $T_2(x, y, z) = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

2. Find the inverse of matrix $B = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$.

3. Compute the determinant of $B$: $\text{det}(B) = 1(1 \cdot 1 - 0 \cdot 2) - 0(2 \cdot 1 - 0 \cdot 0) + 2(2 \cdot 2 - 1 \cdot 0) = 1 - 0 + 8 = 9$

4. Find the adjugate of matrix $B$ and multiply by $\frac{1}{\text{det}(B)}$ to get $B^{-1}$.

5. The inverse matrix $B^{-1}$ is calculated as: $B^{-1} = \frac{1}{9} \begin{bmatrix} 1 & 2 & -4 \\ 2 & 4 & -2 \\ 2 & 1 & 1 \end{bmatrix}$

6. Therefore, the inverse transformation $T_2^{-1}(x, y, z)$ is: $T_2^{-1}(x, y, z) = \left( \frac{1}{9}x + \frac{2}{9}y - \frac{4}{9}z, \frac{2}{9}x + \frac{4}{9}y - \frac{2}{9}z, \frac{2}{9}x + \frac{1}{9}y + \frac{1}{9}z \right)$

Would you like more details on any step or have any questions?

Related Questions:

1. What is the geometric interpretation of a linear transformation and its inverse?
2. How do you find the inverse of a 3x3 matrix using cofactor expansion?
3. What properties must a matrix have to be invertible?
4. How can linear transformations be used in computer graphics?
5. What is the significance of the determinant in linear transformations?

Tip: The determinant of a matrix not only helps in finding its inverse but also provides information about the scaling factor of the transformation.