FM - Core Pure 1 - 7) Linear transformations

Question 1

Equation for matrix transformations

Answer 1

|^a_c^b_d||^x_y| = |^ax+by_cx+dy|

Question 2

Matrices representing reflections in the x-axis and y-axis, and their invariant lines and points

Answer 2

Reflection in x-axis:

• |¹₀⁰_-1|
• y = 0 | all points on it

Reflection in y-axis:

• |^-1₀⁰₁|
• x = 0 | all points on it

Question 3

Matrices representing reflections in the line y = x and y = -x, and their invariant lines and points

Answer 3

Reflection in y = x:

• |⁰₁¹₀|
• y = x | and all points on it

Reflection in y = -x:

• |⁰_-1^-1₀|
• y = -x | and all points on it

Question 4

Matrix representing a rotation of angle θ anticlockwise about the origin and its invariant points and lines

Answer 4

• |^cos(θ)_sin(θ)^-sin(θ)_cos(θ)|
• The origin | only when θ = 180° → any line passing through the origin

Question 5

Transformation represented by |^a₀⁰_b| when a ≠ b and a = b

Answer 5

• 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)

Question 6

Invariant points and lines of all types of stretches

Answer 6

• Strectch in x-direction only → all points on the y-axis | x = 0 and any horizontal line
• Strectch in y-direction only → all points on the x-axis | y = 0 and any vertical line
• Enlargement → centre of enlargement | and any line passing through it

Question 7

Way to find the area scale factor of a stretch or enlargement and what a negative value shows

Answer 7

• Find determinant of the transformation matrix
• If it’s negative → shape has been reflected

Question 8

Side of a matrix that a transformation matrix goes when multiplying

Answer 8

|Transformation matrix||matrix|

Question 9

Matrices representing reflections in the planes x = 0, y = 0 and z = 0

Answer 9

• 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|

Question 10

Matrices representing rotations of angle θ anticlockwise about the x-axis, y-axis and z-axis

Answer 10

• 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|

Question 11

Way to reverse the transformation by a matrix

Answer 11

Transform by the inverse of that matrix

Question 12

Way to find the invariant points and lines of any 2D transformation

Answer 12

• Points → solve the system for X and Y: |^a_c^b_d||^X_Y| = |^X_Y|
• Lines → solve the system for M and C: |^a_c^b_d||^X_MX+C| = |^X’_MX’+C|