Download Due Date: 10 May 2015

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 ⟶ 𝑊 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 −1
b) [−2 −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: 𝑈 ⟶ 𝑉 be a linear transformation, then:
a. The kernel of 𝑇 is a subspace of 𝑈
b. The kernel of 𝑇 is a subspace of 𝑉
c. The range of 𝑇 is a subspace of 𝑈
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
𝑥
𝑇 −1 ([𝑦])will be equal to:
a. T − 1(x, y) = (x – 2y, − 2x + 3/2y)
b. T − 1(x, y) = (x – 1/2y, − 2x + 5y)
c. T − 1(x, y) = (x – 1/2y, − 2x + 3/2y)
d. T − 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(𝑥, 𝑦) = (𝑥 2 , 𝑦 )is linear?
Hint: check if this transformation preserves additivity.
Solution:
We have 𝑇((𝑥, 𝑦) + (𝑧, 𝑤)) = 𝑇(𝑥 + 𝑧, 𝑦 + 𝑤) =
((𝑥 + 𝑧)2 , 𝑦 + 𝑤) ≠ (𝑥 2 , 𝑦) + (𝑧 2 , 𝑤) = 𝑇(𝑥, 𝑦) +
𝑇(𝑧, 𝑤).
So, T does not preserve additivity. So, T is not linear.
2. Let T(x, y, z) = (3x − 2y + z, 2x − 3y, y − 4z).
Write down the standard matrix of T, and compute
T(2, −1, −1).
Solution:
3
A= [2
0
−2
−3
1
1
0]
−4
T(2, −1, −1) =
3
[2
0
−2
−3
1
1
7
2
0 ] [−1] = [7]
3
−4 −1
3. Find an LU-decomposition of the following matrix:
A=[
1
2
3
]
5
Solution:
1
[
2
𝐴 = 𝐿𝑈
1 0 𝑢1
3
]=[
][
𝑙1 1 0
5
𝑢2
𝑢3 ]
3
4
𝑢1
3
] = [𝑙 𝑢
5
1 1
1
[
2
𝑢2
𝑙1 𝑢2 + 𝑢3 ]
𝑢1 = 1, 𝑢2 = 3, 𝑙1 = 2
𝑢3 = −1
1
So L = [
2
1
0
],U = [
−1
0
3
].
1
Note: factorization is not unique.
Another one: L = [
1
0
] ,U = [
1
0
1
2
3
].
−1
4. Find the singular values of the matrix A if given the
following matrix:
4
A A = [−2
−2
T
0
1
0
1
0] .
1
Solution:
The characteristic equation for the matrix is:
−𝜆3 + 6𝜆2 − 11 𝜆 + 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 = 6𝑥 + 7𝑦
2𝑥 + 3𝑦 ≤ 12
2𝑥 + 𝑦 ≤ 8
𝑥, 𝑦 ≥ 0.
Solution:
Coordinates
(0,0)
(0,4)
(4,0)
(3,2)
𝑍 = 6𝑥 + 7𝑦
0
𝑍 = 28
𝑍 = 24
𝑍 = 32
The maximum value is 𝑍 = 32 at the point (3,2).
5