这个作业是完成人工智能相关的数学题
SIT787 – Mathematics for Artificial Intelligence
for the correct answer.
• This assignment has 100 marks and will contribute 15% of your final score.
• Please solve all the linear systems using Gaussian elimination.
• Please write your answers neatly.
• When you finish, please submit a single PDF file as your final work.
Question 1) Consider this matrxi
A =




1 1 0
−1 0 1
0 0 1
1 1 0




.
For matrix A:
(a) Find row echelon form of A through Gaussian Elimination for solving Ax = b, and based
on the value of a
b =




a
1
1
0




,
determine different states of the system. For what values of a the system is definite, for
what values of a the system is inconsistent, and for what values of a the system is indefinite.
(b) Find rank(A).
(c) What are the pivots?
(d) Is the matrix invertible?
(e) What is dim(N(A))?
[8+3+3+3+3= 20 marks]
SIT787 Math for AI Assignment 2 2020 Tri 1 Page 2 of 3
Now consider
A =


3 −1
1 3
1 1


for questions 2, 3, 4, and 5.
Question 2) Find S = AT A matrix and
(a) Find the characteristic polynomial of S
(b) Find the eigenvalues of S
(c) Find the eigenvectors S.
(d) Are the eigenvectors orthonormal? If they are not, convert them into orthobnormal using
Gram-Schmidt process. Call them −→v1,
−→v2.
(e) Consider λ1 ≥ λ2.Set σ1 =

λ1 and σ2 =

λ2 and make this matrix
D =


σ1 0
0 σ2
0 0

 .
Make a matrix V with the orthonormal columns you obtained corresponding to the order
of eigenvalues V =
−→v1
−→v2

. Show that V is an orthogonal matrix.
[2+2+2+8+3+3= 20 marks]
Question 3) Find T = AAT matrix and
(a) Find the characteristic polynomial of T
(b) Find the eigenvalues of T
(c) Find the eigenvectors T.
(d) Are the eigenvectors orthonormal? If they are not, convert them into orthobnormal using
Gram-Schmidt process. Call them −→x1,
−→x2,
−→x3.
[2+3+3+9+3= 20 marks]
SIT787 Math for AI Assignment 2 2020 Tri 1 Page 3 of 3
Question 4) Consider two orthonormal vectors you obtained in question 2.
(a) Using them we want to make three orthonormal vectors −→u1,
−→u2,
−→u3 such that
−→u1 =
1
σ1
A
−→v1
−→u2 =
1
σ2
A
−→v2
Also, find the third vector −→u3,
−→u3 =


a
b
c


such that −→u3 is a unit vector and orthogonal to −→u1 and −→u2. In other words
||−→u3||2 = 1 and −→u3 ⊥
−→u1 and −→u3 ⊥
−→u2.
Put these three vectors in a matrice U =
−→u1
−→u2
−→u3

.
(b) Show that U is an orthogonal matrix.
(c) Compute UDV T
.
(d) What you can say about UDV T
?
[8+5+5+2=20 marks]
Question 5) Consider the system


3 −1
1 3
1 1



x
y

=


2
4
3

 .
(a) Show that this sytem is inconsistent. Technically, it is called over-determined, as there are
more equations than the number of unknowns. This is the case in reality. The number of
cases is much larger than the number of features.
(b) However, we are interested in a solution that minimizes the distance ||Ax − b||. Let’s call
this best vector wˆ . Find wˆ using
wˆ = (A
T A)
−1A
T
b
(c) Using matrices U, V and
D¯ =
 1
σ1
0 0
0
1
σ2
0



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