3 - linear transformations

Question 1

applying a matrix - a b c d to (x y)

Answer 1

a b
c d
->
ax+by
cx+ dy

Question 2

determining a matrix for a transformation

Answer 2

look at the unit vectors
(1,
0) - i
(0
1) - j
find how they change and what they change to and this is the new matrix

Question 3

rotate theta degrees

Answer 3

cosθ - sinθ
sinθ cosθ

Question 4

rotate 90

Answer 4

0 -1
1 0

Question 5

rotate 180

Answer 5

-1 0
0 -1

Question 6

rotate 270

Answer 6

0 1
-1 0

Question 7

reflection in x axis

Answer 7

1 0
0 -1

Question 8

reflection in y axis

Answer 8

-1 0
0 1

Question 9

reflection in the line y = x

Answer 9

0 1
1 0

Question 10

reflection in the line y = - x

Answer 10

0 -1
-1 0

Question 11

enlargement by scale factor k

Answer 11

k 0
0 k

Question 12

shear x axis invariant

Answer 12

(1 k
0 1

Question 13

shear y axis invariant

Answer 13

1 0
k 1

Question 14

stretch by a in x direction and b in y direction

Answer 14

a 0
0 b

Question 15

det (rotation matrix)

Answer 15

1

Question 16

det ( reflection matrix)

Answer 16

-1

Question 17

det(engagement matrix)

Answer 17

k²

Question 18

how det affects area

Answer 18

area of an image after transformation = det(M) * object

Question 19

combined transformations

Answer 19

to do A then B
we do B(A(x)) = BA x

Question 20

inverse matrices to inverse transformation

Answer 20

if x and y are column vectors
if Ax = y
x = A-1 y

Question 21

3*3 rotation about x axis

Answer 21

1 0 0
0 cosθ -sinθ
0 sinθ cosθ

Question 22

3*3 rotation about y axis

Answer 22

cosθ 0 sinθ
0 1 0
-sinθ 0 cosθ

Question 23

3*3 rotation about z axis

Answer 23

cosθ -sinθ 0
sinθ cosθ 0
0 0 1

Question 24

reflect in x/y/z = 0

Answer 24

1 0 0
0 1 0
0 0 1
(whichever line it is that one is -1 eg for x = 0 the first 1 becomes -1)

Question 25

invariant point

Answer 25

unaffected by a transformation

Question 26

invariant line

Answer 26

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

Question 27

invariant points method

Answer 27

(matrix (u = (u
) v) v)
then multiply and simul eq

Question 28

invariant line method

Answer 28

eg y = 2x
sub v into eq
eg v = 2u
(matrix (u = (u’
) 2u) v’)
multiply to find v’ and u’
then if y = 2x v’ = 2u’ - so sub u’ into the eq and hope it = v’

Question 29

for an image to be coplanar (all points lie in same plane)

Answer 29

det of transformation = 0