Y1, C7 - Linear Transformations Flashcards
What does the determinant of a 2d transformation represent
The scale factor of the area
What does the determinant of a 3d transformation represent
The scale factor of the volume
Can you use a matrix to represent a translation
No
What happens to the origin in any linear transformation
It stays the same (0,0)
What types of transformations can matrices be used to represent
Reflections, rotations, enlargements
How can you multiply four different 2d coordinates by a 2x2 transformation (M) at the same time
Write it as M * (2x4) matrix
With coordinate one being the left column etc.
What 2x2 matrix represents a reflection in y axis
(-1, 0, 0, 1)
Are rotations clockwise or anti-clockwise by default
Anti-clockwise
What is the 2x2 matrix for a rotation x about the origin
(cosx, -sinx, sinx, cosx)
What is the 2x2 matrix for a reflection in y=x
(0, 1, 1, 0)
What are invariant points
Points which are unaffected by transformations
What are invariant lines
When each point on the line transformed to give another point on the same line
What point is always an invariant point
The origin (0,0)
Describe fully the transformation described by (1/root(2), -1/root(2), 1/root(2), 1/root(2))
arccos(1/root(2)) = 45
arcsin(-1/root(2)) = -45
Rotation 45 degrees about origin
Do transformations always have invariant lines
NO
Matrix P = (3, 3, 4, 7), describe (a) invariant points, and (b) invariant lines
a) (3, 3, 4, 7) * (x, y) = (x, y)
–> 3x + 3y = x, therefore y = -2/3 * x
and 4x + 7y = y, therefore y = -2/3 * x
Therefore all lines on point y = -2/3 * x are invariant
b) y = mx + c
(3, 3, 4, 7) * (x, mx + c) = (z, mz + c)
–> 3x + 3(mx + c) = z
and 4x + 7(mx + c) = mz + c
–> 4x + 7(mx+c) = m(3x+3(mx+c)) + c
0 = 3m^2 - 4xm + 3mc - 6c - 4x
0 = x(3m+2)(m-2) + 3c(m-2)
when m = 0, c can take any value
when m = -2/3, c = 0
Therefore invariant lines are y=-2/3 * x and y = 2x+c
What does the matrix (a, 0, 0, b) represent
Stretch of scale factor a parallel to x-axis, stretch of scale factor b parallel to y-axis. If a = b it is an enlargement
If we transform a vector x by matrix A, then matrix B, what is the matrix of the combined matrix
BA
Matrix for a rotation about x-axis in 3d
(1, 0, 0, 0, cosx, -sinx, 0, sinx, cosx)
Matrix for a rotation about y-axis in 3d
(cosx, 0, sinx, 0, 1, 0, -sinx, 0, cosx)
Matrix for a rotation about z-axis in 3d
(cosx, -sinx, 0, sinx, cosx, 0, 0, 0, 1)
What is important to remember about the matrix transformation for a rotation about the y axis in 3d compared to rotations about x and z axis
The rotation is anticlockwise relative to the positive y-axis therefore the y matrix has its -sinx and sinx swapped
Suppose x and y are column vectors, if Ax = y, what does x equal
x = A^-1 * y
Suppose x and y are column vectors, if BAx = y, what does x equal
x = A^-1 * B^-1 * y
