Alg1231W6T4 - Reflections

Sunday, 2 September 2018

6:56 PM

Machine generated alternative text:
Figure 1: Reflection in the y-axis 
-4 
given by 
O 
-2 
4 
Consider the transformation T : R 
which is reflection in the y-axis The map T can be represented by the matrix 
1, 
-2 
y 
1 
2 
y 
-1 
o 
Then: 
y 
iii) If T(v) 
1 
, then x 
6 
10 
and T(w) 
19 
1 
, then v 
-4 
3 
—9 
O and y 
, then 
v) If we reflect the line y = 7m + 6 in the y-axis we get the line y 
6 
mm + b where 
O 
and b 
6 
1 
Recall: the Maple notation for the vector 
is < 
2

 

Machine generated alternative text:
Figure: Reflection in the x-axis 
-2 
which is reflection in the x-axis. Then 
Consider the transformation R : IR2 IR2 
—4 
—1 
4 
2 
-2 
-4 
O 
x 
i) The map R can be represented by the matrix 
iii) If 
y 
iv) If we reflect the line y 
Recall: 
then x 
_4 
O and y 
9m + 3 in the x-axis we get the line y 
mat + b where 
and b 
1 
• the Maple notation for the matrix 
3 
1 
• the Maple notation for the vector 
2 
2 
is 
4 
1,2 
112

 

Machine generated alternative text:
Figure: Reflection in the origin 
-4 
-2 
2 
O 
2 
x 
4 
-4 
which is reflection in the origin [0, 0]. 
Consider the transformation S : IR2 IR 
Then 
and the matrix that represents S is 
In terms of the transformations T and R , which are reflections in the y-axis and x-axis respectively, we have [select all that apply] 
S(v) = 
S(v) 
S(v) — 
S(v) — 
S(v) 
Ifwe reflect the line y = 7x + 5 in the origin, then we get the line y 
ma: + b where

 

Machine generated alternative text:
Figure: Reflection in the line y 
6 
-2 
4 
2 
o 
-2 
2 
4 
6 
An important linear transformation in calculus is the reflection U in the line y = x, since this corresponds to taking the inverse of a function. 
For example 
and in general 
The matrix that represents U is 
This matrix has the property that .42 
O fib, as do all 2 x 2 reflection matrices. 
Note: please use the Maple notation for a 2 x 2 when entering .42 
Ifwe reflect the line y = 6x + 9 in the line y = x we get the line y = mm + b where 
and 
b 
-32 
Recall: 
1 
• the Maple notation for the matrix 
3 
1 
• the Maple notation for the vector 
2 
O 
112 >,< 314 
2 
is 
4 
1,2

 

Machine generated alternative text:
Consider the linear transformation 
Then 
and 
1 
Recall: the Maple notation for the vector 
is < 1,2 
2 
—3 
5 
4 
5 
4 
5 
3 
5 
1 
By drawing a picture; and perhaps some experimentation, we can determine that V is a reflection about the line 
(Make sure to enter your answer as an 
equation in Maple syntax; for example 
If we reflect the line y = 8x + 7 using V, we get the line y 
ma: + b where 
28/29 
o 
and 
b 
-35/29

 

 

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