MATH2206 Prob Stat/25.Sept.2015
Weekly Review 3
This week we had three hours, during which we nished Chapter 4 and started Chapter 5,
which introduces a very important notion called random variable. We dened the meaning
of the distribution, the mean and t

Math 2207 Linear Algebra, Tutorial 3
October 5, 2015
Sections 1.8 & 1.9
1. Let T : R3 7 R3 be the transformation that reflects each vector x = (x1 , x2 , x3 ) through the
plane x3 = 0 onto T (x) = (x1 , x2 , x3 ).
(a) Show that T is a linear transformatio

Math 2207 Linear Algebra, Tutorial 10
November 23, 2015
Section 5.3 Diagonalization
1 4 2
1. Diagonalize the matrix A = 3 4 0 with = 1, 2, 3.
3 1 3
4 0 0
2. Discuss why A = 1 4 0 is not diagonalizable.
0 0 5
3. A is a 5 5 matrix with two eigenvalues. One

Math 2207 Linear Algebra, Tutorial 3
October 5, 2015
Sections 1.8 & 1.9
1. Let T : R3 7 R3 be the transformation that reflects each vector x = (x1 , x2 , x3 ) through the
plane x3 = 0 onto T (x) = (x1 , x2 , x3 ).
(a) Show that T is a linear transformatio

Math 2207 Linear Algebra, Tutorial 8
November 9, 2015
Sections 4.3 Linearly Independent Sets; Bases
1. Assume that A
2 4
A = 2 6
3 8
is row equivalent to
2 4
1
3 1 , B = 0
2 3
0
B. Find bases for NulA and ColA.
0 6 5
2 5 3 .
0 0 0
2. Find a basis for the

Math 2207 Linear Algebra
Tutorial 2
2015/09/21
Section 1.5
1.
2.
3.
4.
(a)(i) Since the coefficient matrix A has three pivot positions, every row must contain a pivot.
Thus, Ax = b has a solution for every possible b.
(c)(ii) The fact that A has two pivot

Math 2207 Linear Algebra, Tutorial 1
September 14, 2015
Section 1.1
1. Mark each statement True or False.
a. An inconsistent system has more than one solution.
b. Two linear systems are equivalent if they have the same solution set.
2. Consider a linear s

Math 2207 Linear Algebra, Tutorial 7
November 2, 2015
Section 4.1 Vector Spaces and Subspaces
x1
1. Let G be the set of all vectors of the form x2 R3 : x1 = x2 x3 .
x3
Is G a subspace of R3 ?
6
12
Sol. No. Counter-example: v = 2 and 2v = 4 . Then v

Math 2207 Linear Algebra, Tutorial 4
October 12, 2015
Sections 1.8 & 1.9
1. If f : X Y and g : Y Z be functions. Then, g f : X Z.
(a) If g f is one-to-one(f is applied first.), then g is one-to-one.
(b) If g f is one-to-one(f is applied first.), then f is

Math 2207 Linear Algebra, Tutorial 8
Nov 16, 2015
Section 4.7 Change of Basis
cfw_
1. Let B = b1 , b2 and C = cfw_
c1 , c2 be bases for a vector space V , and suppose b1 =
c1 +4
c2
and b2 = 5
c1 3
c2 .
(a) Find the change-of-coordinates matrix from B t

Math 2207 Linear Algebra, Tutorial 4
October 12, 2015
Sections 1.8 & 1.9
1. If f : X Y and g : Y Z be functions. Then, g f : X Z.
(a) If g f is one-to-one(f is applied first.), then g is one-to-one.
(b) If g f is one-to-one(f is applied first.), then f is

Math 2207 Linear Algebra, Tutorial 11
November 30, 2015
Section 6.1 orthogonal complements
1
0
1
0
1. If V is the subspace spanned by
0 and 1
1
0
plement V .
find a basis for the orthogonal com
1 1 0 1
Sol. Let B =
. Then V = Row(B), so by Theorem 3

Math 2207 Linear Algebra, Tutorial 8
November 9, 2015
Sections 4.3 Linearly Independent Sets; Bases
1. Assume that A
2 4
A = 2 6
3 8
is row equivalent to
2 4
1
3 1 , B = 0
2 3
0
B. Find bases for NulA and ColA.
0 6 5
2 5 3 .
0 0 0
Sol. For NulA, from the

Math 2207 Linear Algebra, Tutorial 1
September 14, 2015
Section 1.1
1. Mark each statement True or False.
a. An inconsistent system has more than one solution.
b. Two linear systems are equivalent if they have the same solution set.
Sol.
(a) False. Incons

Math 2207 Linear Algebra, Tutorial 11
November 30, 2015
Section 6.1 orthogonal complements
1
0
1
0
1. If V is the subspace spanned by
0 and 1
1
0
find a basis for the orthogonal complement V .
2. Determine each of the statement True or False. Justif

Math 2207 Linear Algebra, Tutorial 7
November 2, 2015
Section 4.1 Vector Spaces and Subspaces
x1
1. Let G be the set of all vectors of the form x2 R3 : x1 = x2 x3 .
x3
Is G a subspace of R3 ?
3m 2
2. Let H be the set of all vectors of the form 2m + 4n

MATH2206 Prob Stat/30.Oct.2015
Weekly Review 8
Because of the mid-term test, we had only 1.5 hours this week.
The rst problem we discussed this week is known as the goodness-of-fit test, which is a
-test to test how good a hypothesised distribution can d

MATH2206 Prob Stat/6.Nov.2015
Weekly Review 9
This week we had three hours and we finished Chapter 11 and almost finished Chapter 12.
In the last review we used the 2 -distribution for testing goodness-of-fit and for testing independence or homogeneity of

MATH2206 Prob Stat/9.Oct.2015
Weekly Review 5
This week we had three hours, finished Chapters 6 and 7 and almost finished Chapter 8.
Last week I explained how to get probabilities from the standard normal table, which
has two pages in the textbook. This w

MATH2206 Prob Stat/27.Nov.2015
Weekly Review 12
Done! We had three hours this week and we finished all topics that I would like to discuss
in this course! Hurray!
Last week we introduced the test statistic T of the Wilcoxon rank-sum test for two independe

MATH2206 Prob Stat/13.Nov.2015
Weekly Review 10
This week we had three hours, and we nished ANOVA and started linear regression.
In the last review we introduced the test called ANOVA, which is used to test whether
several means are all the same or not. F

MATH2206 Prob Stat/20.Nov.2015
Weekly Review 11
We had three hours this week and we concluded our discussion on linear regression and
correlation and started the last chapter, which is devoted to nonparametric tests.
Last week we introduced the simple lin

MATH2206 Prob Stat/2.Oct.2015
Weekly Review 4
In the three hours of this week, we introduced the hypergeometric, the geometric, the
negative binomial, the Poisson, the uniform and the normal (Gaussian) distribution, covered
up to Section 6.3. We also spen

MATH2206 Prob Stat/23.Oct.2015
Weekly Review 7
This week we had three hours, during which we talked about hypothesis testing procedures when the null hypotheses are H0: p = po , H0: 1 = 2 and H0: p1 = p2 . We have also
seen the formulae for confidence int

MATH2206 Prob Stat/11.Sept.2015
Weekly Review 1
Welcome! Starting from this week, you can expect to get a review every week. Last week
we got one hour and this week we had three hours. In these four hours we discussed dierent
ways to summarize data, such

MATH2206 Prob Stat/18.Sept.2015
Weekly Review 2
This week we had three normal lecture hours, during which we illustrated more counting
techniques and introduced formally the mathematical denition of probability and conditional probability. Some results ha

MATH2206 Prob Stat/16.Oct.2015
Weekly Review 6
This week (perhaps the most important week in this course) we started our discussion on
one of the central questions in statistical analysis: hypothesis testing. What I explained
this week (and next week) is