FM - Core Pure 1 - 7) Linear transformations Flashcards
Equation for matrix transformations
|acbd||xy| = |ax+bycx+dy|
Matrices representing reflections in the x-axis and y-axis, and their invariant lines
Reflection in x-axis:
- |100-1|
- y = 0
Reflection in y-axis:
- |-1001|
- x = 0
Matrices representing reflections in the line y = x and y = -x, and their invariant lines
Reflection in y = x:
- |0110|
- y = x
Reflection in y = -x:
- |0-1-10|
- y = -x
Matrix representing a rotation of angle θ anticlockwise about the origin and its invariant point
- |cos(θ)sin(θ)-sin(θ)cos(θ)|
- The origin
Transformation represented by |a00b| when a ≠ b and a = b
- When a ≠ b → stretch by scale factor a in the x-direction and stretch by scale factor b in the y-direction
- When a = b → enlargement by scale factor a (or b)
Invariant points and lines of all types of stretches
- Strectch in x-direction only → points on the y-axis and the line x = 0
- Strectch in y-direction only → points on the x-axis and the line y = 0
- Enlargement → only the origin and no lines
Way to find the area scale factor of a stretch or enlargement
Find determinant of the transformation matrix
Side of a matrix that a transformation matrix goes when multiplying
|Transformation matrix||matrix|
Matrices representing reflections in the planes x = 0, y = 0 and z = 0
- Reflection in x = 0:
|-1 0 0|
|0 1 0|
|0 0 1| - Reflection in y = 0:
|1 0 0|
|0 -1 0|
|0 0 1| - Reflection in z = 0:
|1 0 0|
|0 1 0|
|0 0 -1|
Matrices representing rotations of angle θ anticlockwise about the x-axis, y-axis and z-axis
- Rotation about x-axis:
|1 0 0|
|0 cos(θ) -sin(θ)|
|0 sin(θ) cos(θ)| - Rotation about y-axis:
|cos(θ) 0 sin(θ)|
|0 1 0|
|-sin(θ) 0 cos(θ)| - Rotation about z-axis:
|cos(θ) -sin(θ) 0|
|sin(θ) cos(θ) 0|
|0 0 1|
Way to reverse the transformation by a matrix
Transform by the inverse of that matrix