2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 9
(for the week of 3-7 December 2012)
Revision Points:
Properties of eigenvalues and eigenvectors
Diagonalisation of matrices
1. Let A =
32
.
3 2
(a) Show that the characteristic polynomial of A i
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 8
(for the week of 26-30 November 2012)
Revision Points:
Similar matrices
Eigenvalues and eigenvectors
1. Dene L : P3 P3 by L(p(x) = xp (x) + p (x).
(a) Find the matrix A representing L with respe
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 7
(for the week of 19-23 November 2012)
Revision Points:
Kernel and range of linear transformations
Matrix representation of linear transformations
1. For the linear transformation L : R3 R2 dened
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 6
(for the week of 12-16 November 2012)
Revision Points:
Row space, column space and null space
Rank, nullity and related theorems
Denitions and examples of linear transformations
1. By applying
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 5
(for the week of 5-9 November 2012)
Revision Points:
Basis
Dimension
Change of basis
1. If V is a subspace of a vector space W , is it true that a basis of V
(a) must be a subset of any given b
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 4
(for the week of 22-26 October 2012)
Revision Points:
Linear combination
Span and spanning set
Linear dependence and independence
1. As vectors in P3 , can x2 4x +6 be written as a linear combi
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 3
(for the week of 15-19 October 2012)
Revision Points:
The adjoint of a matrix and its applications
Cramers rule
Examples and non-examples of vector space
Subspace
1. Let A = (aij ) be an n n m
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 2
(for the week of 8-12 October 2012)
Revision Points:
Conditions for invertibility of a matrix (without determinants)
Minors and cofactors
Determinants and properties
1. Let A be a square matrix
2012-13 First Semester
MATH 1111 Linear Algebra
Tutorial 1
(for the week of 1-5 October 2012)
Revision Points:
Solutions to linear systems in relation to reduced row echelon form
Operations on matrices and their properties
Row operations and elementary
2012-13 First Semester
MATH 1111 Linear Algebra
Test 3
Outline of Solutions
1. (a) 2I3
(b) Span(1, 2)T )
(c) 1
(d) 1, 4, 6
(e) (2 )3
2. (a) For p(x) = a + bx, we have
1
1
p(x) dx =
0
0
1
(a + bx) dx = ax + bx2
2
1
=a+
0
b
2
and p(0) = a. Hence L(p(x) = (a
201213 First Semester
MATH 1111 Linear Algebra
Test 3 Report
A.
Statistics
Score Distribution
Score Range
30 39
40 49
50 59
60 69
70 79
80 89
90 99
No of students
6
12
32
23
10
3
3
Question
1
2
3
4
5
Mean score
17.53
15.15
16.37
3.12
6.12
Item statistics
2012-13 First Semester
MATH 1111 Linear Algebra
Test 2
Outline of Solutions
1. (a) (There are many possible answers.)
(b) (There are many possible answers.)
(c) 0
(d) 6
(e)
10
01
2
2. We compute the reduced row echelon form of A (which we denote by U ) by
201213 First Semester
MATH 1111 Linear Algebra
Test 2 Report
A.
Statistics
Score Distribution
Score Range
1019
2029
3039
4049
5059
6069
7079
8089
9099
No of students
3
6
9
15
20
17
16
3
2
Item statistics
Question
1
2
3
4
5
Mean score
13.43
14.47
12.62
7.7
2012-13 First Semester
MATH 1111 Linear Algebra
Test 1
Outline of Solutions
10
01
1. (a)
0
(b)
1
4
1
2
1
8
(c) 3
(d) Innitely many
(e) 120
10 0
2. (a) (F) A = 0 1 0 is a counterexample.
0 0 1
(b) (F) A 2 does not
1
0
(c) (F) When A =
0
make sense since A
201213 First Semester
MATH 1111 Linear Algebra
Test 1 Report
A.
Statistics
Score Distribution
Score Range
20 29
30 39
40 49
50 59
60 69
70 79
80 89
No of students
9
17
26
21
11
4
2
Item statistics
Question
1
2
3
4
5
Mean score
15.73
9.11
12.37
2.51
7.38
P
201213 First Semester
MATH 1111 Linear Algebra
Chapter 6: Eigenvalues
Coverage of Chapter 6:
Only Section 6.1 and Section 6.3 will be covered.
Skip The Exponential of a Matrix in Section 6.3.
A.
Examples for Motivation
1.
Evaluate the following.
100
(a)
201213 First Semester
MATH 1111 Linear Algebra
Chapter 4: Linear Transformations
Coverage of Chapter 4:
The entire chapter will be covered.
A.
Definitions and Examples
(Ref: Section 4.1)
1.
Let V and W be vector spaces. A linear transformation from V to W
201213 First Semester
MATH 1111 Linear Algebra
Chapter 3: Vector Spaces
Coverage of Chapter 3:
Skip the subsection The Vector Space C ( n 1) [a, b] in Section 3.3 (Pages 135137).
A.
Definitions and Examples
(Ref: Section 3.1)
1.
Consider
the set of real n
201213 First Semester
MATH 1111 Linear Algebra
Chapter 1: Matrices and Systems of Equations
Coverage of Chapter 1:
Skip Application 3 in Section 1.4.
Skip Triangular Factorisation in Section 1.5.
A.
Solving Equations
1.
We are all familiar with solving eq
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 5
Outline of Solutions
1. (a) The characteristic polynomial of A is
det(A I ) =
3
5
= 2 2 8 = ( + 2)( 4)
1
1
while the characteristic polynomial of B is
4
0
0
det(B I ) = 2 1
= (4 )2 (1 ).
0
5
3
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 5
Not to be handed in
1. (Easy! ) Let
A=
35
1 1
4 00
and B = 2 1 0 .
5 34
For each matrix,
(a) nd its characteristic polynomial;
(b) nd all its eigenvalues;
(c) nd the eigenspace corresponding to
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 4
Due Date: 23rd November, 2012 (Friday), 5:00 p.m.
Please read the remarks at the beginning of Assignments 1 and 2.
All linear transformations are assumed to be between nite dimensional vector
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 3
Outline of Solutions
1. To prove that the vectors 1 x2 , x + 2, x2 span P3 , we should prove that any arbitrary
vector ax2 + bx + c in P3 where a, b, c R can be written as a linear combination o
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 3
Due Date: 6th November, 2012 (Tuesday), 5:00 p.m.
Please read the remarks at the beginning of Assignments 1 and 2.
1. (Easy! ) Show that 1 x2 , x + 2 and x2 span P3 .
2. (Easy! ) Are 1 x2 , x +
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 2
Outline of Solutions
1. (a)
12
34
= 1 4 3 2 = 2
(b) Expanding along the third row, we have
123
345
012
= 1
13
12
+2
35
34
= 0.
(c) Let A denote the 4 4 matrix in the question. Observe that the
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 2
Due Date: 22nd October, 2012 (Monday), 5:00 p.m.
Write down your name (as on your HKU student card) and student number. Write neatly
on A4-sized paper and show your steps.
Put your assignment
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 1
Due Date: 8th October, 2012 (Monday), 5:00 p.m.
Write down your name (as on your HKU student card) and student number. Write neatly
on A4-sized paper and show your steps.
Put your assignment i
2012-13 First Semester
MATH 1111 Linear Algebra
Assignment 1
Outline of solutions
Remark. All questions in this assignment can be solved without using determinants. Although
it is acceptable if you use determinants properly (after we have established the