Homework 2 (Due date: 18th February)
Let k1 and k2 be two real numbers. Let A, B, and C be matrices.
Prove the following statements:
(1) A + B = B + A,
(2) (A + B) + C = A + (B + C),
(3) (AB)C = A(BC),
(4) A(B + C) = AB + AC,
(5) (A + B)C = AC + BC,
(6)
2016 MATH1030
Classwork 10
1. Let
a
e
A=
0
r
b c d
f 0 0
g h i
s t 0
Write down A(2|1) and A(4|3).
2. Let A M44 . Express det B in terms of det A by the following row or column
operation (in below letter C stands for column operations)
R R
2
(a) A 1
B
3R
2016 MATH1030
Classwork 7 Solution
Notation: Suppose A is a square matrix. A0 = In , A1 = A, A2 = AA, A3 = AAA,
A4 = AAAA etc. Obviously Ak+1 = Ak A = AAk .
1. (Skip this question if you have learned basic matrix operations in high school)
(a) Let
1 0 1
1
2016 MATH1030
Classwork 9
1. Let V = R3 , Show that
x
W = cfw_ y V |2x + 3y + 4z = 1
z
is not a subspace of V .
2. Let V = R2 . Let
x
W =cfw_
V |x y.
y
Show that W is not a subspace of V by stating clearing which condition it fails and
give an explic
2016 MATH1030
Classwork 7
Notation: Suppose A is a square matrix. A0 = In , A1 = A, A2 = AA, A3 = AAA,
A4 = AAAA etc. Obviously Ak+1 = Ak A = AAk .
1. (Skip this question if you have learned basic matrix operations in high school)
(a) Let
1 0 1
1 1 2
A =
2016 MATH1030
Classwork 10
The solution is for reference only, it may contain typos and errors. Read at
your own risk.
1. Let
a b
e f
A=
0 g
r s
c d
0 0
h i
t 0
Write down A(2|1) and A(4|3).
Answer.
b
A(2|1) = g
s
c d
a
h i , A(4|3) = e
t 0
0
b
f
g
d
0 .
2016 MATH1030
Classwork 6
Suggestion:
Beginner: finish Q1-Q9.
Veteran: skip the computational questions that you think are easy and finish the rest of
the CW.
1. Let V = Rm . Let S = cfw_v1 , v2 , v3 , v4 , v5 and T = cfw_v1 , v2 , v3 , v4 . Suppose
2v1
2016 MATH1030
Classwork 9 Solution
1. Let V = R3 , Show that
x
W = cfw_ y V |2x + 3y + 4z = 1
z
is not a subspace of V .
Answer. Since 2(0) + 3(0) + 4(0) 6= 1, 0
/W
Therefore, W is not a subspace of V .
2. Let V = R2 . Let
x
W =cfw_
V |x y.
y
Show t
2016 MATH1030
Classwork 2
The solution is for reference only
Please write down you name, student id and session (math1030A, B, C)
on the answer sheets
1. Determine if the following matrices are of RREF.
If yes, copy the matrix on the answer sheets, circle
Homework 1 (Due date: 1st Februrary)
Section 1.1, Problem 3, 9; Section 1.2, Problem 1, 6, 8, 9, 10, 11, 12.
For problem 6 of section 1.2, please also nd vectors c, v1 , ., vl such
that the set of solutions is of the form
cfw_c + k1 v1 + .kl vl |k1 , ., k
2016 MATH1030
Classwork 4
Important remark: In below V always denotes a vector space. If you feel uncomfortable of abstract vector space, you can assume V = Rk .
1. For the following system of linear equation
x1 + 2x2 + 12x3 + 15x4 + 3x5 + 42x6
x1
2x3 3x
2016 MATH1030
Classwork 5
Suggestion:
Beginner: Q1-Q12
Veteran: Q2, Q7-Q17
This is a very long and difficult classwork. We do not expect you to finish it during the
class. Go home and finish all the unfinished questions.
1. Let
1
1
1
1
S = cfw_v1 , v2
Suggestion for beginner: Finish Q1, Q2, Q3ab.
1. Solve the following system of linear equations. State your steps clearly.
(a)
2x1 x2 = 1
3x1 + 2x2 = 12
(b)
x1 x2 x3 = 0
2x1 + 3x2 3x3 = 11
x1 + 3x2 + 6x3 = 1
2. For the following system of linear equation
2016 MATH1010
Classwork 3
Midterm Information: Midterm I: Oct 3, Mon, 7:30 to 9:15 pm LSK LT5.
Lecture 1 to Lecture 6 (i.e., up to the definition of vector spaces).
Review notes and sample questions will be posted later.
Suggestion: Finish Q1-4, 7, 9a, 10
2016 MATH1010
Classwork 3 Solution
The solution is for reference only. It may contains typos. Read at your
own risk
Midterm Information: Midterm I: Oct 3, Mon, 7:30 to 9:15 pm LSK LT5.
Lecture 1 to Lecture 6 (i.e., up to the definition of vector spaces).
2016 MATH1030
Classwork 2
Please write down you name, student id and session (math1030A, B, C)
on the answer sheets
1. Determine if the following matrices are of RREF.
If yes, copy the matrix on the answer sheets, circle the leading ones and write down
th
2016 MATH1030
Classwork 5
Please write down you name, student id and session (math1030D, E, F) on
the answer sheets
The solution is for reference only. It may contain typos and error. Read at
your own risk.
1. Let
1
1
1
1
1
2
3
S = cfw_v1 , v2 , v3 =
2016 MATH1030
Classwork 4 Solution
The solution is for reference only. It may contains typos. Read at your
own risk
Important remark: In below V always denotes a vector space. If you feel uncomfortable of abstract vector space, you can assume V = Rk .
1.
2016 MATH1030
Classwork 6
Please write down your name, student id and session (math1030A, B, C) on
the answer sheets.
The solution is for reference only, it may contain typos and errors. Read at
your own risk.
Suggestion:
Beginner: finish Q1-Q9.
Veteran: