Linear Transformations

Question 1

How are they represented

Answer 1

( a b) (x)

c d) (y

Question 2

How to know if it is a linear transformation

Answer 2

If the output is (ax +by)

(cx + dy)

Question 3

Facts

Answer 3

Cannot do translations
Can change the number of dimensions
The origin is always unaffected

Question 4

How to find the matrix that represents the translation

Answer 4

Put in form:
(ax + by)
(cx + dy)
Remove x and y for the output

Question 5

Unit vector representing the x direction

Answer 5

( 1 )

0

Question 6

Unit vector representing the y direction

Answer 6

( 0 )

1

Question 7

Rotation θ about the origin

Answer 7

( cos θ -sin θ)

sin θ cos θ

Question 8

What are all rotations?

Answer 8

Measured in degrees, anti-clockwise about the origin

Question 9

Invariant point

Answer 9

Unaffected by a transformation

Either only the origin or a whole line of them passing through the origin

Question 10

Invariant line

Answer 10

Where each point of the line is transformed to another point on the same line

Question 11

Finding the matrix to represent a transformation that isn’t a rotation

Answer 11

1. Take the column vectors for x and y
2. Work out what they will be after the transformation
3. Write that as a 2x2 matrix with x on the left and y on the right

Question 12

How to find the area scale factor of a transformation?

Answer 12

The absolute value of the determinant of the transformation matrix

Question 13

Matrix ( a 0)

( 0 b)

Answer 13

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

Question 14

What is the combined transformation if you transform by matrix A and then by matrix B

Answer 14

BA

Question 15

How to go from image to point?

Answer 15

Apply the inverse of the transformation matrix in front of the inverse

Show the steps of multiplying left and right by inverse

Question 16

x unit vector 3D

Answer 16

(1)
(0)
(0)

Question 17

y unit vector 3D

Answer 17

(0)
(1)
(0)

Question 18

z unit vector 3D

Answer 18

(0)
(0)
(1)

Question 19

3D reflections

Answer 19

Again see where each point transforms to and form a matrix

It will always be in a plane x/y/z = 0

Question 20

3d rotation θ about the x-axis

Answer 20

(1 0 0)
(0 cosθ -sinθ)
0 sinθ cosθ)

Question 21

3d rotation θ about the y-axis

Answer 21

(cosθ 0 sinθ)
(0 1 0)
(-sinθ 0 cosθ)

Question 22

3d rotation θ about the z-axis

Answer 22

(cosθ -sinθ 0)
(sinθ cosθ 0)
(0 0 1)

Question 23

What must you do after applying a matrix to a point?

Answer 23

Write as a coordinate

Question 24

Finding invariant points

Answer 24

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

Question 25

Finding invariant lines

Answer 25

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