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```CCI Math251 Linear Algebra
2015
Assignment 4
Due Date: 10 May 2015
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Determine whether the statement is true or false:
1. A linear transformation preserves the operations of
vector addition and scalar multiplication. (True).
2. If the linear transformation π : π βΆ π is both one-toone and onto, then it is an isomorphism. (True).
3. If we interchange two rows in an identity matrix, then
it will not have an LU-decomposition. (True).
4. Every square matrix has an LU-decomposition. (False).
5. LU decompositions are unique. (False).
6. Simplex method is an iterative procedure for solving
LPP in finite number of steps. (True).
7. The solution set of a system of linear equation is
bounded if it can be enclosed by a circle. (True).
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1
2
For Each Question, Choose the Correct Answer from
the Multiple-Choice List.
1. One of the following matrices has no LU-decomposition
1 0
a) [0 1
0 0
0
0]
1
2
1 β1
b) [β2 β1 2 ]
2
1
0
0 1 0
c) [1 0 0]
0 0 1
1
d) [2
4
0 0
3 0]
1 2
2. Let π: π βΆ π be a linear transformation, then:
a. The kernel of π is a subspace of π
b. The kernel of π is a subspace of π
c. The range of π is a subspace of π
d. None.
3. Graphical method can be applied to solve LPP when
there are only:
a. One variable.
b. Two variables.
c. Three variables.
d. None.
3 1
4. Let π: β2 βΆ β2 be the multiplication by [
] then
4 2
π₯
π β1 ([π¦])will be equal to:
a. T β 1(x, y) = (x β 2y, β 2x + 3/2y)
b. T β 1(x, y) = (x β 1/2y, β 2x + 5y)
c. T β 1(x, y) = (x β 1/2y, β 2x + 3/2y)
d. T β 1(x, y) = (x β 1/2y, 2x + 3/2y)
Solve the following questions:
1. Determine whether the function
CCI Math251 Linear Algebra
2015
π: β2 βΆ β2 , where π(π₯, π¦) = (π₯ 2 , π¦ )is linear?
Hint: check if this transformation preserves additivity.
Solution:
We have π((π₯, π¦) + (π§, π€)) = π(π₯ + π§, π¦ + π€) =
((π₯ + π§)2 , π¦ + π€) β  (π₯ 2 , π¦) + (π§ 2 , π€) = π(π₯, π¦) +
π(π§, π€).
So, T does not preserve additivity. So, T is not linear.
2. Let T(x, y, z) = (3x β 2y + z, 2x β 3y, y β 4z).
Write down the standard matrix of T, and compute
T(2, β1, β1).
Solution:
3
A= [2
0
β2
β3
1
1
0]
β4
T(2, β1, β1) =
3
[2
0
β2
β3
1
1
7
2
0 ] [β1] = [7]
3
β4 β1
3. Find an LU-decomposition of the following matrix:
A=[
1
2
3
]
5
Solution:
1
[
2
π΄ = πΏπ
1 0 π’1
3
]=[
][
π1 1 0
5
π’2
π’3 ]
3
4
π’1
3
] = [π π’
5
1 1
1
[
2
π’2
π1 π’2 + π’3 ]
π’1 = 1, π’2 = 3, π1 = 2
π’3 = β1
1
So L = [
2
1
0
],U = [
β1
0
3
].
1
Note: factorization is not unique.
Another one: L = [
1
0
] ,U = [
1
0
1
2
3
].
β1
4. Find the singular values of the matrix A if given the
following matrix:
4
A A = [β2
β2
T
0
1
0
1
0] .
1
Solution:
The characteristic equation for the matrix is:
βπ3 + 6π2 β 11 π + 6.
The eigenvalues are 1,2 and 3.
Then the singular values of A are 1, β2,and β3.
CCI Math251 Linear Algebra
2015
5. Using graphical method to solve the Linear
Programming Problem.
Maximize
Subject to
π = 6π₯ + 7π¦
2π₯ + 3π¦ β€ 12
2π₯ + π¦ β€ 8
π₯, π¦ β₯ 0.
Solution:
Coordinates
(0,0)
(0,4)
(4,0)
(3,2)
π = 6π₯ + 7π¦
0
π = 28
π = 24
π = 32
The maximum value is π = 32 at the point (3,2).
5
```
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