Alg1231W6T3 - Linear transformations through pictures

Sunday, 2 September 2018

6:29 PM

Untitled picture.png Machine generated alternative text:
Figure 1. A colourful house 
-2 
c 
A 
2 
Suppose OE 
7 
8 
Under the transformation S(x) 
3 
2 
Nx where N is 
(t) and = 
we find that S(OA) 
It follows that 
S(öö) = 
S(OE) - 
vectoræ, = 7/2, (2) = 9' 
1 
Recall: the Maple notation for the vector 
is < 1,2 
2 

Untitled picture.png Machine generated alternative text:
Figure 1. The same colourful house 
-2 
2 
Suppose OE 
we find that T(OA) 
7 
8 
3 
2 
Under the transformation T(x) 
Mx , where M is 
2 
-1 
c 
A 
4 
2 
and T(OB) 
-1 
4 
T(OC) — 
T(OD) — 
T(OE) — 
vectoræ, = 2, (2) = 3}) 
Vector(2, = 7/4, (2) = 
1 
Recall: the Maple notation for the vector 
2 
Untitled picture.png Machine generated alternative text:
with(LtnearÅ lgebra) 
03 
Multiply(N, (1, 1)) 
4 
Multiply(N, 1.5)) 
Multiply N, 
2. 
450000000000000 
2 
9 
2
Untitled picture.png Machine generated alternative text:
Figure 1. The same colourful house 
-2 
2 
Suppose OE 
we find that T(OA) 
7 
8 
3 
2 
Under the transformation T(x) 
Mx , where M is 
2 
-1 
c 
A 
4 
2 
and T(OB) 
-1 
4 
T(OC) — 
T(OD) — 
T(OE) — 
vectoræ, = 2, (2) = 3}) 
Vector(2, = 7/4, (2) = 
1 
Recall: the Maple notation for the vector 
2 

Untitled picture.png Again suppose that OE 
Under the transformation R(x) 
Figure 1 The same colourful house 
Lx , where L is 
we find that R( 
R(OC) 
R(OD) — 
R(OE) 
and R(OB) 
vector(2, = 5., (2) = 2.5 
so 
Untitled picture.png Machine generated alternative text:
with(LtnearÅ lgebra) 
Multiply(N, (1, 1)) 
2 
Multiply(N, 1.5)) 
Multiply N, 
4 
41 
Untitled picture.png with(LtnearÅ lgebra) 
13 
-12 
Multiply(N, (1, 1)) 
4 
Multiply(N, 1.5)) 
Multiply N, 
250000000000000 
43 
17
Untitled picture.png Again suppose that OE 
Under the transformation R(x) 
Figure 1 The same colourful house 
Lx , where L is 
we find that R( 
R(OC) 
R(OD) — 
R(OE) 
and R(OB) 
vector(2, = 5., (2) = 2.5 
so 

Untitled picture.png Which of the following is the image of the house under the transformation R? For convenience, these diagrams have been scaled dowm 
Untitled picture.png with(LtnearÅ lgebra) 
13 
-12 
Multiply(N, (1, 1)) 
4 
Multiply(N, 1.5)) 
Multiply N, 
250000000000000 
43 
17
Untitled picture.png Which of the following is the image of the house under the transformation R? For convenience, these diagrams have been scaled dowm 

Untitled picture.png Consider the matrix "renovation" U that will transform our house in figure 1 into the house in Figure 2 below 
Figure 1. The usual house, without 
the marked points 
Figure 2_ The transformed house 

Untitled picture.png 2 
o 
+2 
1 
2 
4 
5 
Playing with the app above (or otherwise), we see that the matrix representation for this linear transformation would be K 
This transformation would send v 
Recall: the Maple notation for the matrix 
112 >,< 314
Untitled picture.png 2 
o 
+2 
1 
2 
4 
5 
Playing with the app above (or otherwise), we see that the matrix representation for this linear transformation would be K 
This transformation would send v 
Recall: the Maple notation for the matrix 
112 >,< 314 

Untitled picture.png Figure 1 _ A logo 
Figure 2 The transformed logo 
A designer wants to arrange a transformation R to take the image in Figure 1 to the image in Figure 2. Unfortunately she didn't take a first year maths course at UNSWI but we can help her 
by looking fora linear 
O transformation to do the job; i.e. something of the form 
ax + by 
The definition of the map R tells us that 
and R( 
Untitled picture.png with(LtnearÅ lgebra) 
Multiply(N, ( 1 , 4)) 
21 
02 
-6
Untitled picture.png Figure 1 _ A logo 
Figure 2 The transformed logo 
A designer wants to arrange a transformation R to take the image in Figure 1 to the image in Figure 2. Unfortunately she didn't take a first year maths course at UNSWI but we can help her 
by looking fora linear 
O transformation to do the job; i.e. something of the form 
ax + by 
The definition of the map R tells us that 
and R( 

Untitled picture.png and the diagram tells us 
Note: the Maple notation for the vector 
Hence; after a little bit of algebra; we deduce that 
1,2 > 
and 
satisfy the equation [select all that apply] 
2 
To check our result; we note that if x + y 
5X2 - 2XY+2Y2 = 9 
(X - + (2X+Y)2 
5X2 +2XY+2Y2 = 1. 
1 then
Untitled picture.png and the diagram tells us 
Note: the Maple notation for the vector 
Hence; after a little bit of algebra; we deduce that 
1,2 > 
and 
satisfy the equation [select all that apply] 
2 
To check our result; we note that if x + y 
5X2 - 2XY+2Y2 = 9 
(X - + (2X+Y)2 
5X2 +2XY+2Y2 = 1. 
1 then

 

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