Complex number z = a + bi, a, b R, i =
1.
a = real part of z
b = imaginary part of z.
Benet: Fundamental Theorem of Algebra (FTA): Every
polynomial equation has a solution in the complex numbers.
C = cfw_a + bi | a, b R is the set of complex numbers.
The

Denition: Let V and W be vector spaces. A linear transformation T : V W is invertible if there is a linear transformation
T 1 : W V such that T 1 T = 1v (identity transformation on
V ) and T T 1 = 1w .
Facts: (1) (S T )1 = T 1 S 1
(2) (T 1)1 = T
Theorem:

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 1
Summer 2014
Work on the course material and problems below. You should be prepared to receive instruction
about this assignment in the 1st tutorial, wh

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 6
Summer 2014
Work on the course material and problems below. Quiz 6 is based on this Assignment 6 and the
relevant text readings and lecture notes. See

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 2
Summer 2014
Work on the course material and problems below. You should be prepared to receive instruction
about this assignment in your tutorial in Wee

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 3
Summer 2014
Work on the course material and problems below. Quiz 3 is based on this assignment and the
relevant text readings and lecture notes. Quiz 3

University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Midterm Test #1
MATA37-CALCULUS II FOR MATHEMATICAL SCIENCES
Examiner: R. Grinnell
Date: May 31, 2006
Duration: 110 minutes
PRINT the following:
SouurnoNS
FAMILY N

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 8
Summer 2014
Work on the course material and problems below. Quiz 8 is based on this Assignment 8 and the
relevant text readings and lecture notes.
All

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 4
Summer 2014
Work on the course material and problems below. Quiz 4 is based on this assignment and the
relevant text readings and lecture notes. Quiz 4

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 5
Summer 2014
Work on the course material and problems below. Quiz 5 is based on this assignment and the
relevant text readings and lecture notes. Quiz 5

Denition: Let V and W be vector spaces (over F ). A function
T : V W is called a linear tansformation from V to W if,
for all u, v V and F , we have
(i) T (u + v) = T (u) + T (v)
(preservation of vector addition)
(ii) T ( v) = T (v)
(preservation of scala

(i) The determinant of a 1 by 1 matrix a is a.
(ii) Suppose a denition is provided for a n1 by n1 determinant.
Dene
a11 a12
a21 a22
det
an1 an2
a1n
a2n
=
ann
n
a1j c1j = a11 c11+a12 c12+ +a1n c1n
j=1
where cij is the cofactor of aij .
Recall
Let A =

Carefully tear this page off the rest of your exam.
U NIVERSITY OF T ORONTO S CARBOROUGH
MATA37H3 : Calculus for Mathematical Sciences II
R EFERENCE S HEET
The (natural) logarithm and exponential functions
(1) The (natural) logarithm function is dened:
x

Denition: A eld F is a set on which two operations + and
(addition and multiplication) are dened, so that, for each x, y in
F , there are unique elements x + y and x y in F (closed under +
and ) for which the following conditions hold for all elements a,

Denition: A basis for a vector space V is a linearly independent
subset of V that generates V .
If is a basis for V , we say that the elements of form a basis
for V .
This means (1) is a linearly independent subset of V .
(2) the set of elements of span V

U NIVERSITY OF T ORONTO S CARBOROUGH
MATA37H3 : Calculus for Mathematical Sciences II
MIDTERM EXAMINATION # 2
March 9, 2012
Duration 2 hours
Aids: none
NAME (PRINT):
Last/Surname
First/Given Name
KEY
STUDENT NO:
TUTORIAL:
Tutorial section
(Number or Sched

Elementary Properties of vector spaces.
Proposition: Cancellation Law for Vector Addition.
If u, v, w V , a vector space, are such that u + w = v + w, then
u = v.
Corollary 1: The zero vector (condition iii) is unique.
Corollary 2: The inverse of a vector

U NIVERSITY OF T ORONTO S CARBOROUGH
MATA37H3 : Calculus for Mathematical Sciences II
MIDTERM EXAMINATION # 1
January 30, 2012
Duration 2 hours
Aids: none
NAME (PRINT):
Last/Surname
SOLUTION KEY
First/Given Name
STUDENT NO:
TUTORIAL:
Tutorial section
(Num

University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MATA37
Assignment 7
Summer 2014
Work on the course material and problems below. Quiz 7 is based on this Assignment 7 and the
relevant text readings and lecture notes.
All