(7) Linear Transformations

How do linear transformations work?

- The only terms involved are x and y, there are no other constants

- They transform the origin onto itself (therefore, the origin is always an invariant point)

- The linear transformation T:(X Y) > (aX + bY + cX + dY) can be represented by the matrix M (a b c d) [the matrix is a 2x2 matrix]

What matrix represents a reflection in the y-axis? And state the lines that are invariant upon this transformation

(-1 0)

(0 1)

Invariant lines are x=0 and y=k for any k

What matrix represents a reflection in the x-axis? And state the lines that are invariant upon this transformation

(1 0)

(0 -1)

Invariant lines are y=0 and x=k for any k

What matrix represents a reflection in the line y=x? And state the lines which are invariant upon this transformation

(0 1)

(1 0)

Invariant lines are y=-x and y=-x+k for any k

What matrix represents a reflection in the line y=-x? And state the lines which are invariant upon this transformation

(0 -1)

(-1 0)

Invariant lines are y=x and y=x+k for any k

What matrix represents a stretch parallel to the x-axis? And state the lines that are invariant upon this transformation

(a 0

(0 1), for any a

The invariant line is x=0 (the y-axis)

What matrix represents a stretch parallel to the y axis? And state the lines which are invariant upon this transformation

(1 0

(0 a) , for any a

Invariant line is y=0 (the x axis)

What is the formula for the area of a transformed image, T' ?

T' = T x Det (T)

NOTE: Det (T) > 0

How do successive transformations work?

A matrix with two or more transformations is represented by the last matrix on the left hand side multiplied by any successive matrices on the right hand side, in order.

For a transformation Q, represented by Mat (q) followed by matrix P, represented by Mat (p), the matrix representing these transformations is pq

What matrix represents a reflection in the plane x=0 axis for a 3D transformation?

(-1 0 0)

(0 1 0)

(0 0 1)

What matrix represents a reflection in the plane y=0 axis for a 3D transformation?

(1 0 0)

(0 -1 0)

(0 0 1)

What matrix represents a reflection in the plane z=0 axis for a 3D transformation?

(1 0 0)

(0 1 0)

(0 0 -1)

What matrix represents a rotation of angle X in the plane x=0 axis for a 3D transformation?

(1 0 0)

(0 cos(X) -sin(X))

(0 sin(X) cos(X))

What matrix represents a rotation of angle X in the plane y=0 axis for a 3D transformation?

(cos(X) 0 sin(X))

(0 1 0)

(-sin(X) 0 cos(X))

What matrix represents a rotation of angle X in the plane z=0 axis for a 3D transformation?

(cos(X) -sin(X) 0)

(sin(X) cos(X) 0)

(0 0 1)

How can a matrix transformation be cancelled out?

By multiplying it by it's inverse Matrix (M^-1)

What does the determinant represent?

Area scale factor