Alg1231W6T2 - Linear maps
Sunday, 2 September 2018
5:07 PM
Created with Microsoft OneNote 2016.
![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)](Alg1231W6T2%20-%20Linear%20maps_files/image001.png)
![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](Alg1231W6T2%20-%20Linear%20maps_files/image002.png)
![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](Alg1231W6T2%20-%20Linear%20maps_files/image003.png)
![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](Alg1231W6T2%20-%20Linear%20maps_files/image005.png)
