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Math113 Linear Algebra: Final Examination
Dept of Math, HKUST, Spring 1998 Name: ID No. Tutor: Section:
Problem No. 1 (100 pts) No. 2 (90 pts) No. 3 (90 pts) No. 4 (70 pts) No. 5 (150 pts) Total (500 pts) Score 1. Consider the linear transformation T : R4
Math 113, L2, Midterm Exam, Spring 2000
1. Answer the following questions. Provide a counter example for each false question (3 points for each true question and 5 points for each false question). b (1). The equation A~ has a solution for any ~ 2 R m if A
Math 113, L4, Solutions of Midterm Exam, Fall 2000
Date:
30 October 2000
Time:
7:00p.m.| 8:00p.m. Name Student Number Tutorial Section Score
Venue:
LTC
1. Answer the following questions. Provide a counter example for each false question (3 points for each
Math 113, L5, Solutions of Midterm Exam, Fall 2000
Date:
30 October 2000
Time:
3:55p.m.| 4:55p.m. Name Student Number Tutorial Section Score
Venue:
RM2465
1. Answer the following questions. Provide a counter example for each false question (3 points for e
Chapter 5: Orthogonality
April 30, 2009
Week 13-14
1
Inner product
Geometric concepts of length, distance, angle, and orthogonality, which are well-known in R2 and R3 , can be dened in Rn . These concepts provide powerful geometric tools to solve many app
Chapter 4: Eigenvalues and Eigenvectors
April 17, 2009
Week 11-12
1
Eigenvalues and eigenvectors
0 0 . . . n x1 x2 . . . xn ,
If a linear transformation T : Rn Rn has the form x1 1 0 x2 0 2 T . = . . . . . . . . . . xn 0 0
then we can easily see that T
Chapter 3: Vector Spaces
March 16, 2009
Lecture 16
1
Vector spaces
A vector space is a non-empty set V of objects, called vectors, on which are dened two operations, called addition and scalar multiplication: for any vectors u, v in V , the sum u + v is i
Chapter 2: Matrices and Determinants
March 16, 2009
1
Linear transformations
Lecture 7
Denition 1.1. Let X and Y be nonempty sets. A function from X to Y is a rule f : X Y such that each element x in X is assigned a unique element y in Y , written as y =
(b) cd ab ab c d = cb da = (ad bc) = ab . cd ab . cd
4
Determinants
Lecture 12 Let T : R2 R2 be a linear transformation with the standard matrix A= a11 a12 . a21 a22
a11 a12 a21 a22
Part (c) follows from Part (a). As for Part (d), we have = ac bd = (ad bc
Shrikant Patnaik
4 Determinants
Lecture 12 Let T : R2 R2 be a linear transformation with the standard matrix a11 a12 . A= a21 a22 Recall that the determinant of a 2 2 matrix det
a11 a12 a21 a22
Digitally signed by Shrikant Patnaik DN: cn=Shrikant Patnaik,
4
Determinants
Lecture 12 Let T : R2 R2 be a linear transformation with the standard matrix a11 a12 . A= a21 a22
Theorem 4.2. Let A = [aij ] be an n n matrix. The (i, j )-cofactor of A (1 i, j n) is the number Cij = (1)i+j det Aij . Then det A = a11 C11 +