Linear transformations

Question 1

What is a linear transformations?

Answer 1

All elements in the matrix are in the form (Ax + By). There are no constants.

Question 2

How can a linear transformation be defined?

Answer 2

By its effect on (1 0) and (0 1).

If M = (a b c d), then (a b c d)(1 0) = (a c), and (a b c d)(0 1) = (b d)

Question 3

What is an invariant point?

Answer 3

A point which is mapped to itself under a given transformation.

Question 4

What is an invariant line?

Answer 4

A line which is mapped to itself under a given transformation.

Question 5

Give the matrix, invariant points, and invariant lines for a reflection in the y-axis.

Answer 5

(-1 0 0 1)
Points: All points on x = 0
Lines: x = 0, y = k

Question 6

Give the matrix, invariant points, and invariant lines for a reflection in the x-axis.

Answer 6

(1 0 0 -1)
Points: All points on y = 0
Lines: y = 0, x = k

Question 7

Give the matrix, invariant points, and invariant lines for a reflection in the line y = x.

Answer 7

(0 1 1 0)
Points: All points on the line y =x
Lines: y = -x, y = -x +k

Question 8

Give the matrix, invariant points, and invariant lines for a reflection in the line y = -x

Answer 8

(0 -1 -1 0)
Points: All points on the line y = -x
Lines: y = -x, y = x + k

Question 9

Give the matrix, invariant points, and invariant lines for a rotation through angle w anticlockwise about the origin.

Matrix is given in formula booklet, invariants are not.

Answer 9

Matrix: (cosw -sinw sinw cosw)
Points: (0,0)
Lines: If w ≠ 180, none
If w = 180, any line passing through origin

Question 10

Give the matrix, invariant points, and invariant lines for a stretch of scale factor a parallel to y = 0 and a stretch of scale factor b parallel to x = 0.

If a = b, it is an enlargement with scale factor a.

Answer 10

(a 0 0 b)
Points: (0,0)
Lines: (y = 0, x = 0)

Question 11

Give the matrix, invariant points, and invariant lines for a stretch of scale factor a parallel to y = 0 only.

Answer 11

(a 0 0 1)
Points: All points on x = 0
Lines: y = k (any line parallel to y = 0)

Question 12

Give the matrix, invariant points, and invariant lines for a stretch of scale factor b parallel to x = 0 only.

Answer 12

(1 0 0 b)
Points: All points on y = 0
Lines: x = 0 (any line parallel to x = 0)

Question 13

How can the area of an image be found?

Answer 13

Aimage = Aobject x |DetM|

Question 14

In terms of transformations, what does matrix PQ represent?

Answer 14

Transformation Q followed by transformation P.

Question 15

How is a reflection in the plane x = 0 represented?

Answer 15

(-1 0 0)
(0 1 0)
(0 0 1)

Question 16

How is a reflection in the plane y = 0 represented?

Answer 16

(1 0 0)
(0 -1 0)
(0 0 1)

Question 17

How is a reflection in the plane z = 0 represented?

Answer 17

(1 0 0)
(0 1 0)
(0 0 -1)

Question 18

How is a rotation anticlockwise about x-axis, angle w represented?

Answer 18

(1 0 0)
(0 cosw -sinw)
(0 sinw cosw)

Question 19

How is a rotation anticlockwise about y-axis, angle w represented?

Answer 19

(cosw 0 sinw)
(0 1 0)
(-sinw 0 cos0)

Question 20

How is a rotation anticlockwise about z-axis, angle w represented?

Answer 20

(cosw -sinw 0)
(sinw cosw 0)
(0 0 1)

Question 21

What effect does an inverse matrix have?

Answer 21

Reverses the effect of the transformation.

Question 22

How can invariant points of (a b c d) be found?

Answer 22

(a b)(X) = (X)
(c d)(Y) = (Y)

Question 23

How can invariant lines of (a b c d) be found?

Answer 23

(a b)(—-X—-) = (—-X—-)
(c d)(MX + C) = (MX’ + C)