MAT-703
SUMMARY NOTE ON COMPLEX NUMBERS
T. Zakon
An overview of the evolution of number systems
Starting with counting numbers, our number system evolved gradually for a variety of purposes, both
computational and algebraic. At each stage of the evolution
MAT- 703
Summary Note on Subspaces, Spans, Basis, Dimension
T. Zakon
CONCEPTS AND DEFINITIONS
v1 , v2 , vm is a vector of the form w = c1v1 + c2 v2 + + cm vm where
the ci are scalars ; in other words w is a weighted sum of the vi , with weights ci .
1.
MAT-703
T. Zakon
STUDY GUIDE FOR TEST I
September, 2016
Note: You should expect test questions to involve definitions, interpretation and comprehension, some
proofs, as well as questions involving computations. Material covered involves Practice sheet 1,
MAT-703
STUDY GUIDE FOR TEST II
October, 2016
T. Zakon
Complex Numbers
Definitions (be able to state):
What is a complex number? What are the real part and imaginary part
of a complex number? What is the absolute value and the polar angle of a complex num
MAT-703
T. Zakon
ASSIGNMENT 3
Complex Arithmetic
REFERENCES: TEXTBOOK PP 234 249; Summary note on Complex Numbers (to be posted on
Omnivox)
HAND IN questions 3(h), (i), 4, 5(b), 6, 7(a), (b), 8(a), (b), 9
Complex Numbers in Rectangular form
1.
Specify the
MAT- 703
Assignment 4
T. Zakon
Subspaces, basis and dimension
See posted Summary Document on Subspaces. It includes definitions and outline of procedures relevant to the
problems below. You may use the online Matrix Reducer for matrices involving numeric
201-NYC-05 Fall 2016
Homework
M. Hamel
August 17, 2016
Homework: Homework will be assigned weekly. There will be three types of homework assigned:
WeBWorK, written and collected assignments, and suggested exercises. You may not have all
types of homework
MAT NYC
ASSIGNMENT 3
1. A square (n x n) matrix P is called idempotent if P 2 P. (For example the zero matrix On
and the identity matrix I n are idempotent.) Show that:
1 1 1 0
1 1 1
a)
and
are idempotent.
,
2 1 1
0 0 1 0
b) If P is idempotent, so is I
MAT NYC
Linear Algebra
Solutions
Course Package
Jean-Francois Deslandes
Marianopolis College
Fall 2015
Section 1
Vector algebra
Problem 1.1
Consider the vectors, , and , shown below.
u v
w
Find the graphical result of the following vector operations. Appr
Section 1
Vector algebra
Problem 1
The structure of a methane (CH4) molecule is such that the carbon atom is positioned at the
center of the four hydrogen atoms. The four hydrogen atoms are at positions
A(0, 0, 0), B(1, 0, 1), C(1, 1, 0), D(0, 1, 1).
a)
MAT NYC
Linear Algebra
Course Package
Jean-Francois Deslandes
Marianopolis College
Fall 2015
2
Differential Calculus
Mat NYC
Fall 2015
Instructor
Office
E-mail
Phone
Jean-Francois Deslandes
I-322
[email protected]
514-931-8792 (ext. 412)
My sch
Section 3
Linear systems and Gaussian elimination
Problem 1
Using Gaussian elimination, reduce the following system to its echelon form. Identify all free and
leading variables and write its general solution.
3 x 3 y z 2t 10
x 4 y 5 z 3t 5
5 x 5 y 9 z 8t
Section 4
Matrix algebra
Problem 1
Using index notation, demonstrate the left-distributivity property:
C(A+ B) = CA + CB.
Demonstration done in notes
Problem 2
1
1
2
6
A quadrilateral has its vertices at the coordinate points A , B , C , D .
1
3
2
Section 6
Vector spaces and subspaces
Problem 1
Define the following concepts :
a)
b)
c)
d)
S is a subspace
The subspace S = spancfw_ u , v , w
Linear independence of vectors u , v , w .
The vectors u , v , w are a basis for a subspace S.
Problem 2
Section 5
Determinants and its applications
Problem 1
2
4
1
A 3 1 13
Consider the matrix
4
1
k2
Compute the determinant of A. What values of k will lead to 1 solution, no solution or infinitely
many solutions
Solution
1
det A 3
4
a)
2
1
1
4
1
13 0
k2
Section 2
Lines and planes
Problem 1
Consider the plane whose equation is 2 x 3 y 6 z 12 . Find a unit vector that is normal to the
plane.
Solution
2 3 6
, ,
7 7 7
Problem 2
The vertices of a triangle are A(1, 2, 1), B(2, 3, 1) and C(4, 2, 5).
a)
Find t
201 NYC
Homework 3
Problem 1.16
Let
u 3 b 2 a
, where and are perpendicular vectors.
a
b
c) Find the length of the vector
Problem 1.17
Let us assume that
u
and
v
;
a u
are unit vectors that are separated by a 30 degree angle. Reduce each of the
following
201 NYC
Homework 2
Problem 1.13
Consider the triangle show below. Use a projection to find the coordinates of point N if it is the point on
the segment AB that is nearest to point C.
Problem 1.15
Determine whether or not the following statements are true.
201 NYC
Homework 1
Problem *
Consider a triangle whose vertices are
and
C
. Assume that
Q
of the distance between
segment
that
P
,
A B
,
is the point that is 2/3
A
and the midpoint of
BC . In the same manner, assume
is the point that is 2/3 of the distanc
MAT-NYC
Lab 4: Polar Form of Complex Numbers
1. Convert the following numbers to rectangular form.
a) 3 cos 4 + i sin 4
b) 5 cis
c) cos 7 + i sin 7
3
3
2
4
4
d) 10 cis 248 (give two decimal places)
2. Convert the following numbers to polar form. Choose t
MAT-NYC
Lab 5: Lines & Planes in R3
1. True or false? (Note: distinct in this context means not coincident)
a) Two distinct planes perpendicular to a given line are parallel to each other.
b) A plane and a line not in the plane either intersect or are par
201-NYC E
1.
Assignment 4
a) A triangle is formed by joining two adjacent corners at the
bottom of a cube to the midpoint of the edge at the top of
the opposite face, as in the picture on the right. Use the dot
product to calculate the internal angles of
Course 2
August 27, 2015
8:20 AM
Nature vs Nurture
Answer might have a big impact on social policy
Where is if worth intervening to help kids and families?
Burt sometimes it is hard to know!
Children active in own development
Inborn qualities elicit
201-NYC
Exercises on (R)REF, rank
1. Decide if each of the following matrices is in RREF, REF (bot not RREF), or neither REF nor
RREF.
1 4 3 0
1 0
a) 0 0
0 0
0 1
b)
0 1 1 7 0 0
9 0 0
d) 0 0 1
0 0 0
0 0 1
2. Reduce the following matrices to RREF.
a)
1 5
201-NYC
Exercises on Number of Solutions
1. Give the general solution to the linear system whose augmented matrix has been reduced to the
following. Give the solution in (i) parametric equation form, and (ii) vector form.
1 2 0 10 0 55
0 0 1 5 0 40
0 0
201-NYC
Exercises on Elimination
1. Solve the following systems of linear equations by Gaussian elimination with back-substitution.
x1
2x2 4x3 + 13x4 = 11
3x1 +
8x2 + 16x3 49x4 = 45
2x2 3x3 + 14x4 = 6
2x1
6x2 14x3 + 35x4 = 44
2. Solve the
3x
3x
a)
x
f
Christopher Turner
Marianopolis College
201-NYC-05 (Linear Algebra I); A15
Addendum to Course Outline: Evaluation Method
Your evaluation in this course will comprise a series of assignments, two tests and a final
exam.
From the course outline, the break-d
COURSE OUTLINE
Linear Algebra I
TERM:
PONDERATION:
DISCIPLINE:
COURSE CREDIT:
PREREQUISITE:
OFFICE HOURS:
Autumn 2015
3-2-3 (class-lab-homework)
Mathematics
2 2/3
Sec. V Math TS or Sec. V
Math SN or equivalent
Posted by your teacher on
Omnivox and on his/