Alg1231W6T2 - Linear maps

Sunday, 2 September 2018

5:07 PM

Untitled picture.png Machine generated alternative text:
A linear map T: R * R 
has the property that 
01)) co 
Hence 
and 
2m 
Recall: the Maple notation for 
is < 
and T( 

Untitled picture.png Machine generated alternative text:
Another linear map S : IR2 R has the property that 
Hence 
and 
and S( 

Untitled picture.png Machine generated alternative text:
Figure 1. A house. 
2 
1 
The image ofthe house in Figure 1 under the linear map R : IR2 R defined by R( 
O 
2 
and R( 
1 
-1 
is (pick one) 
-2 
O The point which gets sent to [1, —2] is the point 
If the tip of the roof is initially [1/2, 3/2] then the image point of this under R would be the point <-0.5, 
O The point which gets sent to [1, O] is the point 
Untitled picture.png Machine generated alternative text:
u (2-4)
Untitled picture.png Machine generated alternative text:
Figure 1. A house. 
2 
1 
The image ofthe house in Figure 1 under the linear map R : IR2 R defined by R( 
O 
2 
and R( 
1 
-1 
is (pick one) 
-2 
O The point which gets sent to [1, —2] is the point 
If the tip of the roof is initially [1/2, 3/2] then the image point of this under R would be the point <-0.5, 
O The point which gets sent to [1, O] is the point 

Untitled picture.png Machine generated alternative text:
LetT:R 
be a function; or map, or transformation, satisfying 
1 
1 
-2 
1 
and T( 
i) We can express 
6 
where [al ,a2, — 
as a linear combination of the standard basis vectors; we can write 
0 
Note: make sure to enter your coefficients inside square brackets (e.g [1 , 2 , 3] 
ii) If the function T is a linear map; then T( 
Note: the Maple notation for the vector 
2 
<-36, 32, 
1,2,3 > 
iii) If 
6 
-35 
32 
14 
could T still be a linear map? No 
Untitled picture.png Machine generated alternative text:
(-4-2, 1): 
-36 
32 
14
Untitled picture.png Machine generated alternative text:
LetT:R 
be a function; or map, or transformation, satisfying 
1 
1 
-2 
1 
and T( 
i) We can express 
6 
where [al ,a2, — 
as a linear combination of the standard basis vectors; we can write 
0 
Note: make sure to enter your coefficients inside square brackets (e.g [1 , 2 , 3] 
ii) If the function T is a linear map; then T( 
Note: the Maple notation for the vector 
2 
<-36, 32, 
1,2,3 > 
iii) If 
6 
-35 
32 
14 
could T still be a linear map? No 

Untitled picture.png Machine generated alternative text:
Figure 1 _ A house 
Figure 2_ Linearly '"renovated" house 
-2 
A linear map S takes the house in Figure 1 to the house in Figure 2. Then 
1 
2 
3 
2 
1 
Note: the Maple notation for the vector 
1,2 > 
2 
Untitled picture.png Machine generated alternative text:
(-4-2, 1): 
-36 
32 
14
Untitled picture.png Machine generated alternative text:
Figure 1 _ A house 
Figure 2_ Linearly '"renovated" house 
-2 
A linear map S takes the house in Figure 1 to the house in Figure 2. Then 
1 
2 
3 
2 
1 
Note: the Maple notation for the vector 
1,2 > 
2 

Untitled picture.png Machine generated alternative text:
It appears this transformation multiplies all areas by a factor of 
02 
0 7/2 
0 5/2 
3 



Untitled picture.png Machine generated alternative text:
Here is the nutritional information for 20 fl oz of Original Coke , Vanilla Coke , Pepsi 
Sodium (mg) and Carbs (g): 
noc 
and Pepsi Next e, represented as nutritional vectors with the respective amount of Calories, 
240 
75 ,nvc 
65 
260 
60 
70 
, np 
250 
55 
70 
and npnr 
100 
100 
25 
The nutritional vector for original coke is noc can be expressed as a linear combination of vectors of the other colas: 
noc = anvc + ßnp + .npN 
where the coefficients are 
Note: this may be a tricky calculation; see hints below 
The main ingredients of cola are Carbonated Water; High Fructose Corn Syrup (HFCS) and Caramel Colour. The nutritional content for these ingredients is in the table below 
Table 1. Ingredients and Nutritional Information 
Calories 
Sodium (mg) 
Carbs (g) 
We can represent this nutritional information by the matrix A: 
Carbonated Water 
(1 fl oz) 
0 
5 
HFCS 
(1 floz) 
25 
5 
25 
20 
38 
Caramel Colour 
(1 fl oz) 
25 
20 
38 
0 
A 
o 
5 
0 
25 
1 
5 
The matrix A defines a linear map from ingredient vectors to nutritional vectors. For example; we can compute the nutritional value of a mixture of 5 fl oz of Carbonated Water, 6 fl oz of 
HFCS and 2fl oz of Caramel Colour as: 
A 
5 
6 
2 
Untitled picture.png Machine generated alternative text:
with(LtnearÅ lgebra) 
u} (260: 60: 70) 
to (250: 70) 
us (100: 100, : 
LinearSolve( (ullu21u3), (240, 75, 65)) 
2 
4 
15
Untitled picture.png Machine generated alternative text:
Here is the nutritional information for 20 fl oz of Original Coke , Vanilla Coke , Pepsi 
Sodium (mg) and Carbs (g): 
noc 
and Pepsi Next e, represented as nutritional vectors with the respective amount of Calories, 
240 
75 ,nvc 
65 
260 
60 
70 
, np 
250 
55 
70 
and npnr 
100 
100 
25 
The nutritional vector for original coke is noc can be expressed as a linear combination of vectors of the other colas: 
noc = anvc + ßnp + .npN 
where the coefficients are 
Note: this may be a tricky calculation; see hints below 
The main ingredients of cola are Carbonated Water; High Fructose Corn Syrup (HFCS) and Caramel Colour. The nutritional content for these ingredients is in the table below 
Table 1. Ingredients and Nutritional Information 
Calories 
Sodium (mg) 
Carbs (g) 
We can represent this nutritional information by the matrix A: 
Carbonated Water 
(1 fl oz) 
0 
5 
HFCS 
(1 floz) 
25 
5 
25 
20 
38 
Caramel Colour 
(1 fl oz) 
25 
20 
38 
0 
A 
o 
5 
0 
25 
1 
5 
The matrix A defines a linear map from ingredient vectors to nutritional vectors. For example; we can compute the nutritional value of a mixture of 5 fl oz of Carbonated Water, 6 fl oz of 
HFCS and 2fl oz of Caramel Colour as: 
A 
5 
6 
2 

Untitled picture.png Machine generated alternative text:
But with the power of linear algebra, we can also go backwards and compute the following valuable trade secrets (see hints below, and give your answers to two significant figures): 
• The ingredient vector for Vanilla Coke is 
• The ingredient vector for Pepsi is 
7.8 
9.9 
.55 
vector(3, = 67, (2) - 
• The ingredient vector for Pepsi Next is 
Vector(3, (2) 
Hints 
• the Maple command for finding the reduced row-echelon form of a matrix M is 
LinearA1gebra [ReducedRowEche10nForm] (M) • 
• the Maple command to compute the inverse N of the matrix M is 
LinearA1gebra [Matrixlnverse] (M) ; 
• the Maple command to compute the product of a matrix N and vector v , evaluated to two significant figures; is 
evalf [2] (LinearAIgebra [Multiply] (N , v) ) ; 

Untitled picture.png Machine generated alternative text:
restart : 
with(LtnearÅ lgebra) 
(260: 60: 70) . 
to (250: 70) 
us (100: 100: 
LinearSolve( (ullu21u3), (240, 75, 65)) 
2 
4 
15 
0 25 25 
1 20 
0 38 
Multiply(Å, (5, 6, 2)) 
200 
71 
106 
evalf( ) , to ) , 2 ) 
6.7 
9.4 
0.61 
19. 
3.8 
o .15
Untitled picture.png Machine generated alternative text:
restart : 
with(LtnearÅ lgebra) 
(260: 60: 70) . 
to (250: 70) 
us (100: 100: 
LinearSolve( (ullu21u3), (240, 75, 65)) 
2 
4 
15 
0 25 25 
1 20 
0 38 
Multiply(Å, (5, 6, 2)) 
200 
71 
106 
evalf( ) , to ) , 2 ) 
6.7 
9.4 
0.61 
19. 
3.8 
o .15

 

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