Y1, C7 - Linear Transformations

Question 1

What does the determinant of a 2d transformation represent

Answer 1

The scale factor of the area

Question 2

What does the determinant of a 3d transformation represent

Answer 2

The scale factor of the volume

Question 3

Can you use a matrix to represent a translation

Answer 3

No

Question 4

What happens to the origin in any linear transformation

Answer 4

It stays the same (0,0)

Question 5

What types of transformations can matrices be used to represent

Answer 5

Reflections, rotations, enlargements

Question 6

How can you multiply four different 2d coordinates by a 2x2 transformation (M) at the same time

Answer 6

Write it as M * (2x4) matrix
With coordinate one being the left column etc.

Question 7

What 2x2 matrix represents a reflection in y axis

Answer 7

(-1, 0, 0, 1)

Question 8

Are rotations clockwise or anti-clockwise by default

Answer 8

Anti-clockwise

Question 9

What is the 2x2 matrix for a rotation x about the origin

Answer 9

(cosx, -sinx, sinx, cosx)

Question 10

What is the 2x2 matrix for a reflection in y=x

Answer 10

(0, 1, 1, 0)

Question 11

What are invariant points

Answer 11

Points which are unaffected by transformations

Question 12

What are invariant lines

Answer 12

When each point on the line transformed to give another point on the same line

Question 13

What point is always an invariant point

Answer 13

The origin (0,0)

Question 14

Describe fully the transformation described by (1/root(2), -1/root(2), 1/root(2), 1/root(2))

Answer 14

arccos(1/root(2)) = 45
arcsin(-1/root(2)) = -45
Rotation 45 degrees about origin

Question 15

Do transformations always have invariant lines

Answer 15

NO

Question 16

Matrix P = (3, 3, 4, 7), describe (a) invariant points, and (b) invariant lines

Answer 16

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

Question 17

What does the matrix (a, 0, 0, b) represent

Answer 17

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

Question 18

If we transform a vector x by matrix A, then matrix B, what is the matrix of the combined matrix

Answer 18

BA

Question 19

Matrix for a rotation about x-axis in 3d

Answer 19

(1, 0, 0, 0, cosx, -sinx, 0, sinx, cosx)

Question 20

Matrix for a rotation about y-axis in 3d

Answer 20

(cosx, 0, sinx, 0, 1, 0, -sinx, 0, cosx)

Question 21

Matrix for a rotation about z-axis in 3d

Answer 21

(cosx, -sinx, 0, sinx, cosx, 0, 0, 0, 1)

Question 22

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

Answer 22

The rotation is anticlockwise relative to the positive y-axis therefore the y matrix has its -sinx and sinx swapped

Question 23

Suppose x and y are column vectors, if Ax = y, what does x equal

Answer 23

x = A^-1 * y

Question 24

Suppose x and y are column vectors, if BAx = y, what does x equal

Answer 24

x = A^-1 * B^-1 * y