Gaussian留学生作业代写、jupyter notebook作业代做、c/c++编程设计作业代写、代做Java，Python实验作业
Homework: Gaussian Elimination 1
Instructor: Prof. Hector D. Ceniceros
General Instructions: You have to integrate all the problems that require coding and/or
numerical computation in a single jupyter notebook. Make sure all your codes have a preamble
which describes purpose of the code, all the input variables, the expected output, your
name, and the date of the last time you modified it. Write your own code, individually. Do
not copy codes! The solutions to the problems that do not require coding must be uploaded
as a single pdf or as part of the jupyter notebook.
1. (a) Implement (write a code) Gaussian Elimination with partial pivoting to solve a
linear system Ax = b, given as output the coefficients of a nonsingular, n × n
matrix A and an n-vector b. Your code should produce the solution x or an error
message (if A is nonsingular) and the information needed for the LU factorization.
(b) Let
5 1 0 2 1
0 4 0 1 2
1 1 4 1 1
0 1 2 6 0(1)
Use your Gaussian Elimination code to solve Ax = b.
(c) Test your Gaussian Elimination code for
5 1 0 2
0 4 0 8
1 1 4 2
0 1 2 2. (2)
2. (a) Let A be an n × n upper or lower triangular matrix. Prove that the determinant
of A is equal to the product of its diagonal entries, i.e. det(A) = a11a22 · · · ann.
(b) Prove that the product of the pivots in the Gaussian Elimination for Ax = b is
equal to the determinant of A up to a sign.
(a) Prove that the product of two n × n lower (upper) triangular matrices is a lower
(upper) triangular matrix.
1All course materials (class lectures and discussions, handouts, homework assignments, examinations, web
materials) and the intellectual content of the course itself are protected by United States Federal Copyright
Law, the California Civil Code. The UC Policy 102.23 expressly prohibits students (and all other persons)
from recording lectures or discussions and from distributing or selling lectures notes and all other course
materials without the prior written permission of the instructor.
1
(b) Let Li be the unit upper triangular matrix that produces the i-th elimination step
in the Gaussian Elimination algorithm, i.e.. (4)
3. Find an LU factorization of the matrix
A =5 1 0
0 4 0
1 1 4
. (5)
4. Find the Choleski factorization A = LLT
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