Construct a 2 × 2 matrix that defines a shear of factor 4 in the y-direction, followed by a scaling of factor 5 in the x-direction, followed by a reflection about the y-axis.
To construct a 2 × 2 matrix that defines the desired transformations, we need to apply the three operations one by one: shear in the y-direction, scaling in the x-direction, and reflection about the y-axis.
1. Shear of factor 4 in the y-direction:
A shear transformation in the y-direction is a linear transformation that moves each point in the y-direction by an amount proportional to its distance from the x-axis. To achieve a shear of factor 4 in the y-direction, we need to move each point vertically by 4 times its y-coordinate.
Let's denote the shear matrix as S_y:
```
S_y = | 1 0 |
| 4 1 |
```
In this matrix, the first row remains the same, but the second row's first element is multiplied by 4.
2. Scaling of factor 5 in the x-direction:
Scaling is a linear transformation that stretches or compresses objects along a particular axis. To achieve a scaling of factor 5 in the x-direction, we need to multiply each point's x-coordinate by 5.
Let's denote the scaling matrix as S_x:
```
S_x = | 5 0 |
| 0 1 |
```
In this matrix, the first element of the first row is multiplied by 5, while the second row remains the same.
3. Reflection about the y-axis:
To reflect a point about the y-axis, we need to change the sign of its x-coordinate while keeping the y-coordinate the same.
Let's denote the reflection matrix as R_y:
```
R_y = | -1 0 |
| 0 1 |
```
In this matrix, the first element of the first row is multiplied by -1, while the second row remains the same.
Now, to get the final matrix, we need to multiply the matrices for each transformation in the reverse order:
```
Final Matrix = R_y * S_x * S_y
```
Performing the matrix multiplication, we get:
```
Final Matrix = | -5 0 | | 1 0 | | 1 0 |
| 0 1 | * | 4 1 | * | 5 0 |
= | -5 0 | | 4 1 | | 20 5 |
```
So, the final 2 × 2 matrix that defines a shear of factor 4 in the y-direction, followed by a scaling of factor 5 in the x-direction, followed by a reflection about the y-axis is:
```
Final Matrix = | -5 0 |
| 20 5 |
```
Note: It's essential to apply the transformations in the correct order because matrix multiplication is not commutative.