Final exam semester 2 2020-2021 | Môn Linear algebra and Algebraic structures - Đại học Sư phạm Kỹ thuật Thành phố Hồ Chí Minh

lOMoAR cPSD| 58759230 HCMC UNIVERSITY OF TECHNOLOGY
FINAL EXAM, SEMESTER 2, 2020-2021 AND
EDUCATION Subject: Linear Algebra and Algebraic Structure HIGH QUALITY TRAINING FACULTY Course code: MATH2001E
------------------------- Number of pages: 02 pages. Duration: 90 minutes. Date of exam:
Materials are allowed during the exam.
Instructions: Show all necessary work, and provide full justification for each answer.
Question 1 (1pt): Solve the following system of equation by using Cramer’s method
Question 2 (1.5pts): Given the matrices and . a) Compute AB and BA. b) Compute det(AB).
Question 3 (1.5 pts): Let .
a) Find LU-factorization of the matrix A.
b) Find bases and dimensions for Null(A).
Question 4 (1.5pts): Let be a subspace of . a)
Find an orthogonal basis of W. b)
Let u=[0,1,2,-1]T . Find the projection
of u on W. Question 5 (2pt) Consider the quadratic form . a)
Let A be the symmetric matrix of the form Q. Find the matrix A. b)
Find a change of variables that transfers Q into a quadraticform with no cross product term. No.: BM1/Q T-PĐBCL-RĐT V Page: 1/1
b) Use the key K to descript the message “****” lOMoAR cPSD| 58759230
Notice: Invigilators should not explain the questions on the exam papers.
Expected Learning Outcomes Questions
[ELO G2.2]: Perform the matrix operations, compute the determinant 1
and the inverse of a square matrix, solve the linear equations, 2
proficiency in using row reduction, apply row reduction to compute
the determinant and the inverse of a matrix, know how to apply to linear models..
[ELO G2.3]: Perform almost problems of vector space: determine a 2b
subspace, a combination of a system of vectors, the (in) dependence of 3
a system of vectors; determine bases, dimension of a space, the
coordinate systems of a vector; compute the change-of-coordinates
matrix; calculate the coordinate of a vector relative to a orthogonal
basis or to a orthonormal basis, find the orthogonal projection of a
vector onto a subspace; proficiency in using Gram-Schmidt process.
[ELO G2.4]: Express a linear transformation as a matrix, find the 4
image and the kernel of a transformation; find the eigenvalues and
eigenvectors of a square matrix; (orthogonally) diagonalize a square
matrix; find the sign of a quadratic forms, transform a quadratic
form into one with no cross-product term.
[ELO G2.5]Construct a binary operations; determine whether a set 5
with (an) operation(s) is a group, a ring or a field; use some
algebraic cryptosystems to encrypt or decrypt messages. May 22, 2018
Head of foundation science group No.: BM1/QT-PĐBCL-RĐTV Page: 1/1