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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 𝑇 : 𝑉 ⟢ π‘Š 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 βˆ’1
b) [βˆ’2 βˆ’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 𝑇: π‘ˆ ⟢ 𝑉 be a linear transformation, then:
a. The kernel of 𝑇 is a subspace of π‘ˆ
b. The kernel of 𝑇 is a subspace of 𝑉
c. The range of 𝑇 is a subspace of π‘ˆ
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 𝑇: ℝ2 ⟢ ℝ2 be the multiplication by [
] then
4 2
π‘₯
𝑇 βˆ’1 ([𝑦])will be equal to:
a. T βˆ’ 1(x, y) = (x – 2y, βˆ’ 2x + 3/2y)
b. T βˆ’ 1(x, y) = (x – 1/2y, βˆ’ 2x + 5y)
c. T βˆ’ 1(x, y) = (x – 1/2y, βˆ’ 2x + 3/2y)
d. T βˆ’ 1(x, y) = (x – 1/2y, 2x + 3/2y)
Solve the following questions:
1. Determine whether the function
CCI Math251 Linear Algebra
2015
𝑇: ℝ2 ⟢ ℝ2 , where 𝑇(π‘₯, 𝑦) = (π‘₯ 2 , 𝑦 )is linear?
Hint: check if this transformation preserves additivity.
Solution:
We have 𝑇((π‘₯, 𝑦) + (𝑧, 𝑀)) = 𝑇(π‘₯ + 𝑧, 𝑦 + 𝑀) =
((π‘₯ + 𝑧)2 , 𝑦 + 𝑀) β‰  (π‘₯ 2 , 𝑦) + (𝑧 2 , 𝑀) = 𝑇(π‘₯, 𝑦) +
𝑇(𝑧, 𝑀).
So, T does not preserve additivity. So, T is not linear.
2. Let T(x, y, z) = (3x βˆ’ 2y + z, 2x βˆ’ 3y, y βˆ’ 4z).
Write down the standard matrix of T, and compute
T(2, βˆ’1, βˆ’1).
Solution:
3
A= [2
0
βˆ’2
βˆ’3
1
1
0]
βˆ’4
T(2, βˆ’1, βˆ’1) =
3
[2
0
βˆ’2
βˆ’3
1
1
7
2
0 ] [βˆ’1] = [7]
3
βˆ’4 βˆ’1
3. Find an LU-decomposition of the following matrix:
A=[
1
2
3
]
5
Solution:
1
[
2
𝐴 = πΏπ‘ˆ
1 0 𝑒1
3
]=[
][
𝑙1 1 0
5
𝑒2
𝑒3 ]
3
4
𝑒1
3
] = [𝑙 𝑒
5
1 1
1
[
2
𝑒2
𝑙1 𝑒2 + 𝑒3 ]
𝑒1 = 1, 𝑒2 = 3, 𝑙1 = 2
𝑒3 = βˆ’1
1
So L = [
2
1
0
],U = [
βˆ’1
0
3
].
1
Note: factorization is not unique.
Another one: L = [
1
0
] ,U = [
1
0
1
2
3
].
βˆ’1
4. Find the singular values of the matrix A if given the
following matrix:
4
A A = [βˆ’2
βˆ’2
T
0
1
0
1
0] .
1
Solution:
The characteristic equation for the matrix is:
βˆ’πœ†3 + 6πœ†2 βˆ’ 11 πœ† + 6.
The eigenvalues are 1,2 and 3.
Then the singular values of A are 1, √2,and √3.
CCI Math251 Linear Algebra
2015
5. Using graphical method to solve the Linear
Programming Problem.
Maximize
Subject to
𝑍 = 6π‘₯ + 7𝑦
2π‘₯ + 3𝑦 ≀ 12
2π‘₯ + 𝑦 ≀ 8
π‘₯, 𝑦 β‰₯ 0.
Solution:
Coordinates
(0,0)
(0,4)
(4,0)
(3,2)
𝑍 = 6π‘₯ + 7𝑦
0
𝑍 = 28
𝑍 = 24
𝑍 = 32
The maximum value is 𝑍 = 32 at the point (3,2).
5
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