这个作业是来自澳洲的关于完成人工智能相关的数学代写，以下是作业具体内容：
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 =a110,
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 −11 31 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 00 σ20 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]
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 =1σ1A−→v1
−→u2 =1σ2A−→v2
Also, find the third vector −→u3,
−→u3 =abc
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 −11 31 1xy=243 .
(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 = (AT A)−1ATb
(c) Using matrices U, V and D¯ = 1σ10 001σ20