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 the matrix that represents the given sequence of transformations (shear, scaling, and reflection), we can start with the standard 2x2 identity matrix and apply each transformation step by step.
Let's define the transformations:
Shear of factor 4 in the y-direction:
A shear transformation in the y-direction can be represented using the following 2x2 matrix:
| 1 4 |
| 0 1 |
Scaling of factor 5 in the x-direction:
A scaling transformation along the x-axis can be represented using the following 2x2 matrix:
| 5 0 |
| 0 1 |
Reflection about the y-axis:
A reflection about the y-axis can be represented using the following 2x2 matrix:
| -1 0 |
| 0 1 |
To combine these transformations, we multiply the matrices in reverse order:
First, we perform the reflection, followed by the scaling, and finally, the shear.
1. Reflection:
| -1 0 |
| 0 1 |
2. Scaling:
| 5 0 |
| 0 1 |
3. Shear:
| 1 4 |
| 0 1 |
To obtain the combined matrix, we multiply these three matrices together:
Combined Matrix = Shear * Scaling * Reflection
Multiplying these matrices, we have:
| 1 4 | | 5 0 | | -1 0 |
| 0 1 | * | 0 1 | * | 0 1 |
Multiplying the first two matrices, we get:
| 1 4 | | 5 0 | | -1 0 |
| 0 1 | * | 0 1 | = | 0 1 |
Performing the matrix multiplication:
| 1*5 + 4*0 1*0 + 4*1 |
| 0*5 + 1*0 0*0 + 1*1 |
Simplifying the multiplication:
| 5 4 |
| 0 1 |
Therefore, the 2x2 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:
| 5 4 |
| 0 1 |