Math 1540 Homework Set 1
Due Date: Sep 22, 2014
1.
(a) Show that the rotation of R2 by a xed angle is a linear transformation.
(b) Find the matrix which corresponds to the rotation of R2 by a xed
angle .
2. Consider a map L : R2 R dened by:
x
y
L
n
m
aij
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MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
Coursework 3 SAMPLE SOLUTIONS
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1. Find all solutions to the following systems of linear equ
Math 1540 Homework Set 5 Solution
1. Estimate the value of e0.1 cos(0.05) using the following two methods:
(a) Estimate e0.1 and cos(0.05) separately, using the 2-nd Taylor series of
the functions f (x) = ex and g(x) = cos x about x = 0. Then, multiply
th
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MATH 1540 Homework Set 1
Solution
1. (a) Let e1 , e2 , . . . , en be the standard basis vectors of Rn :
1
0
0
0
1
0
.
e1 = 0 , e2 = 0 , . . . , en = . .
.
.
.
.
.
0
.
.
0
0
1
Let f1 , f2 , . . . , fm be
THE CHINESE UNIVERSITY OF HONG KONG
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MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
Coursework 4 SAMPLE SOLUTIONS
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1. Let:
0 8 2
5
4 5 7 1
B=
1 4
6
0
2 3 1 9
Compute de
THE CHINESE UNIVERSITY OF HONG KONG
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MATH1540 University Mathematics for Financial Studies 2014-15 Term 1
Coursework 5 SAMPLE SOLUTIONS
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1. Let ~v = h1, 2, 4i, w
~ = h0, 5, 7i. Find:
(a) The
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
Coursework 6 SAMPLE SOLUTIONS
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1. Let L be the line in R3 parameterized by the funct
Math 1540 Homework Set 4 Solution
1. Compute each of the following limits, or show that it does not exist.
2
x y2
(a)
lim sin
(x,y)(0,0)
x+y3
(b)
ln(x2 y + e)
lim
(x,y)(1,2)
(c)
cos(xy)
(x,y)(2,/3) x + y
(d)
x2 y 2
(x,y)(1,1) x y
(e)
x2 xy 2y 2
(x,y)(1,1
Math 1540 Homework Set 3
SAMPLE SOLUTIONS
1. Let L be the line in R3 parameterized by the function:
l(t) = 2, 3, 1 t + 0, 5, 0 ,
t R.
Let P be the point (3, 7, 1) R3 .
(a) Show that P does not lie on L.
(b) Find the point on L which is closest to P .
(c)
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
Coursework 8 SAMPLE SOLUTIONS
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f f 2 f f
,
,
for the following functions:
,
x y x2 x
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1540 University Mathematics for Financial Studies 2015-16 Term 2
Coursework 9 SAMPLE SOLUTIONS
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1. Let F (u, v) = e2u+5v . Suppose u and v are themse
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
Coursework 11 SAMPLE SOLUTIONS
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1. Evaluate
y dA, where R is the region in the xy-pl
THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1540 University Mathematics for Financial Studies 2014-15 Term 1
Coursework 10 SAMPLE SOLUTIONS
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1. Let E be the curve which is the intersection of:
Week 2
Example.
Production Cost per Item
Product
Expenses
A B C
Raw Materials 1 4 1
Labor
2 3 1
Misc.
0 1 2
Product No. Items Produced in one Season
A
200
B
100
C
150
0
10 1 0 1
1 4 1
200
750
@ 2 3 1 A @100A = @850A
0 1 2
150
400
This says that the total
Week 3
Recall:
Every matrix is row equivalent to a matrix in row echelon form. In other
words, one can always transform a given matrix to one in row echelon form by
performing the following row operations:
I. Interchange two rows.
II. Multiply a row by a
THE CHINESE UNIVERSITY OF HONG KONG
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MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
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1. (a) Let A, B be invertible n n matrices. Show that
THE CHINESE UNIVERSITY OF HONG KONG
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MATH1540 University Mathematics for Financial Studies 2015-16 Term 1
Coursework 1 SAMPLE SOLUTIONS
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1. Determine if each of the following maps between ve
Week 4
4.1
Finding the Inverse of a Matrix
Recall:
Theorem. Let A be an n n matrix. The following statements are equivalent:
1. A is invertible.
2. The matrix equation Ax = 0 has x = 0 as its only solution.
3. A is row equivalent to I.
Suppose an n n matr
Math 1540 Homework Set 2
Due Date: Oct 16, 2015
1. Find the determinants of the following matrices:
(a)
10 1
1 2
(b)
6 3 3
0
2 7
9 5 4
(c)
20
0
12
0
2.
7 13 5
6 8 5
1 15 5
0 6 11
(a) Let A be an n n square matrix, a real number. Show that there
exist
Math 1540 Homework Set 3
Due Date: Oct 30, 2015
1. Let L be the line in R3 parameterized by the function:
~l(t) = h2, 3, 1it + h0, 5, 0i,
t R.
Let P be the point (3, 7, 1) R3 .
(a) Show that P does not lie on L.
(b) Find the point on L which is closest to
Math 1540 Homework Set 4
Due Date: Nov 20, 2015
1. Compute each of the following limits, or show that it does not exist.
2
x y2
(a)
lim sin
(x,y)(0,0)
x+y3
(b)
lim
ln(x2 y + e)
(x,y)(1,2)
(c)
cos(xy)
(x,y)(2,/3) x + y
(d)
x2 y 2
(x,y)(1,1) x y
(e)
x2 xy
MATH1540 University Mathematics for Financial Studies
Test 2 SAMPLE SOLUTIONS
Time allowed: 40 mins
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ID:
Marks:
Number of problems: 5. Full marks: 50
Please justify all answers.
1. (10 pts) For each of the following planes in R3 , find an equation of
Week 1
1.1 The Real Vector Space Rn
n 2 N,
~.
v
80 1
9
> x1
>
>
>
>B C
>
< x2
=
B C
n
R = B . C xi 2 R
>@ . A
>
> .
>
>
>
: x
;
n
We call an element of Rn a vector, typically denoted by a symbol of the form
The vector whose entries are all zero is called
Week 7
7.0.1
Parameterization of a line in Rn .
Let O be the origin of Rn . Let L be a line in Rn which passes through a given
point P0 2 Rn , and is parallel to a vector ~ 2 Rn . Each point P on L satises:
v
!
P0 P = t~ ,
v
for some t 2 R. On the other h
Week 9
Denition. We say that a function f in n variables is continuous at P0 2 Domain(f )
if:
lim f (P ) = f (P0 ).
P !P0
Every elementary function (a function constructed from constants, power functions, trigonometric, inverse trigonometric, exponential
Week 8
8.1 Functions in n variables1
Given n 2 N. A real-valued function f in n variables is a map:
f : D ! R,
where the domain D is a subset of Rn .
Example.
f : cfw_(x, y) 2 R2 | x2 + y 2 > 0 ! R
1
f (x, y) = p
.
x2 + y 2
If the domain D of f is not exp
Week 5
5.1 Properties of the Determinant
Let A be an n n matrix.
det A = det A> ,
where A> is the transpose of A, dened by A> = Aji . This follows from
ij
the fact that det A may be computed from the cofactor expansion along any
row or column.
If A is an