7. Matrices & transformations Flashcards
AB != BA
Matrix multiplication is not commutitive
A(BC) = (AB)C
Matrix multiplication is associative
You can only add matrices if…
they have the same m x n
A(B + C) = AB + BC
It is distributive
What is conformable?
M x N order can be multiplied by N x P order to make M x P
A stretch parallel to x axis by scale factor k
K 0
0 1
A stretch parallel to y axis by scale factor k
1 0
0 K
An enlargement at O with scale factor K
K 0
0 K
A rotation 90° anticlockwise about O
0 -1
1 0
A rotation 90° clockwise about O //270° anticlockwise
0 1
-1 0
Leading diagonal rule matricies
Elements in the leading diagonal stay the same
Elements in the opposing diagonal switch signs
A rotation of any angle anticlockwise
cosθ -sinθ
sinθ cosθ
A reflection in the x axis
1 0
0 -1
A reflection in the y axis
-1 0
0 1
Multiplying by
A) Identity Matrix
B) 0/Null Matrix
A) Values stay the same
B) All values become 0
Reflection in the line y = -x
0 -1
-1 0
Shear
A transformation in which all the points are translated parallel to a particular line by a factor which is proportional to the distance of the point from a shear line
Shear parallel to the x axis
1 K
0 1
Shear parallel to the y axis
1 0
K 1
Reflection in the yz plane (x = 0 plane)
-1 0 0
0 1 0
0 0 1
Reflection xz plane (y = 0 plane)
1 0 0
0 -1 0
0 0 1
Reflection xy plane (z= 0 plane)
1 0 0
0 1 0
0 0 -1
Rotation of 180 about the z axis`
-1 0 0
0 -1 0
0 0 1
Invarient Point
A point which is mapped to itself by the transformation
How to find lines of invarient points
- Multiply the matrix by XY and make it equal to XY
- Multiply into two different equations
- If (ax + by = x) = (cx + dy = y) then all the points along the ax + by = x are invariant points
Determinant
ad - bc
If the matix’s determinant is 0 then…
the matrix is singular and has no inverse
Inverse of a 2x2 matrix
1/|M|
(d -b)
(-c a)
AA^-1
Is equal to I
(AB)^-1
A^-1 B^-1
A rotation 180° anticlockwise about O
-1 0
0 -1
`Reflection on the line y = x
0 1
1 0
Applying transformations
if C represents A transformed by B
C = BA
Determinant for rotations, reflections and enlargements
Rotation: detM = 1
Reflection: detM = -1
Enlargement: detM = k²
Reflection in the line y = (tanθ)x
cos 2θ sin 2θ
sin 2θ -cos 2θ
