Linear Transformations Flashcards
How are they represented
( a b) (x)
c d) (y
How to know if it is a linear transformation
If the output is (ax +by)
(cx + dy)
Facts
Cannot do translations
Can change the number of dimensions
The origin is always unaffected
How to find the matrix that represents the translation
Put in form:
(ax + by)
(cx + dy)
Remove x and y for the output
Unit vector representing the x direction
( 1 )
0
Unit vector representing the y direction
( 0 )
1
Rotation θ about the origin
( cos θ -sin θ)
sin θ cos θ
What are all rotations?
Measured in degrees, anti-clockwise about the origin
Invariant point
Unaffected by a transformation
Either only the origin or a whole line of them passing through the origin
Invariant line
Where each point of the line is transformed to another point on the same line
Finding the matrix to represent a transformation that isn’t a rotation
- Take the column vectors for x and y
- Work out what they will be after the transformation
- Write that as a 2x2 matrix with x on the left and y on the right
How to find the area scale factor of a transformation?
The absolute value of the determinant of the transformation matrix
Matrix ( a 0)
( 0 b)
Stretch by scale factor a parallel to the x-axis
Stretch by scale factor b parallel to the y-axis
Enlargement by scale factor a if a = b
About the origin
What is the combined transformation if you transform by matrix A and then by matrix B
BA
How to go from image to point?
Apply the inverse of the transformation matrix in front of the inverse
Show the steps of multiplying left and right by inverse
x unit vector 3D
(1)
(0)
(0)
y unit vector 3D
(0)
(1)
(0)
z unit vector 3D
(0)
(0)
(1)
3D reflections
Again see where each point transforms to and form a matrix
It will always be in a plane x/y/z = 0
3d rotation θ about the x-axis
(1 0 0)
(0 cosθ -sinθ)
0 sinθ cosθ)
3d rotation θ about the y-axis
(cosθ 0 sinθ)
(0 1 0)
(-sinθ 0 cosθ)
3d rotation θ about the z-axis
(cosθ -sinθ 0)
(sinθ cosθ 0)
(0 0 1)
What must you do after applying a matrix to a point?
Write as a coordinate
Finding invariant points
See if ax+by = x and cx+dy = y are equivalent by rearranging for y
If equivalent they represent a line of invariant points
If not the only invariant point is the origin
Finding invariant lines
Use y = mx + c so do M ( x )
(mx + c)
Make the y of answer m(x of answer) + c
Expand and rearrange, group xs and cs together
Solve for m, see what c values make it work and explain
