16 February, 2015
MATH 1107, Section C
Total marks: 30
TEST 2
1. Solve the following systems using an augmented matrix and elementary row operations,
bringing your solution to row reduced echelon form
Free variables and Basis for N(A)
Let A mn be a matrix in reduced row-echelon form. The
solutions to Ax=0 are obtained by setting the free variables to
distinct parameters. The solutions will then be
Test #1 (
/30)
Name (print): _
Part A: Short answer (1 Mark Each)
Question 1:
9 =
a)
b)
Student Number:
c)
1
d)
Question 2: A leading variable of a reduced system can always be assigned as:
a) A basic
05/01/2016
Set Notation
We call a collection of items a set, and it is usually represented using cfw_:
A = cfw_1,4,9,
B = cfw_n2 : n = 1, 2, 3,
Some common sets you should know:
- Empty set
N Natura
Vector Spaces
We have talked about span, but what does a span look like? What else can a vector
look like? What space do they occupy?
Vector Space:
A Vector space is a set of objects (called vectors)
Inverse Matrix
An inverse of an n x n matrix is a matrix B such that AB = BA = I. As we would
like to distinguish that B is the inverse of A, we will call the inverse of A
using an exponent of -1. Tha
Linear Combinations Continued
Given a set of vectors v1, v2, v3, , vn, we call a linear combination:
c1v1+c2v2+c3v3+cnvn
Where c1, c2, c3, , cn can be any set of constants.
Example:
2 0 1
2 1 0
Homogenous System
A Homogenous System is a linear system Ax=b, where b = 0 (the column
vector of all zeros).
A homogenous system is always consistent! (It may be one solution, or
infinitely many solut
Elementary Row Matrices
We will next consider each row operation as a matrix multiplication on the left side.
Row Swap:
Consider the Identity matrix I3 that has the 2nd and 3rd row swapped:
1 0 0
0 1
Augmented Matrix
Recall our convention for using matrices to write systems of linear equations:
Coefficients
Ex: If we have a linear system:
x1 2 x2 3 x3 6
2 x1 x2 x3 7
x1 x2 x3 2
Equations
x1
E1 1
E
Dot Product
We define the dot product between two column tuples v u as the following:
Multiply each corresponding entry of v with u, then sum up each of these
products together.
That is v u v1u1 v2u 2
Identity Matrix
An identity matrix is a matrix that has the same columns as rows (c = r) and
only has 1s along the diagonal, and 0s in every other entry.
A 2x2 identity matrix looks like:
1 0
0 1
A
Test #1 (
/30)
Name (print): _
Part A: Short answer (1 Mark Each)
Question 1:
9 =
a)
b)
Student Number:
1
c)
d)
Question 2: A leading variable of a reduced system can always be assigned as:
a) A basic
CARLETON UNIVERSITY
FINAL
EXAMINATION
December 2010
DURATION:
3
HOURS
School of Mathematics and Statistics
Course Instructor(s) Gang Li, Ranjeeta Mallick
Math 1107 D,E
AUTHORIZED MEMORANDA
NON-PROGRAM
Linear Independence
Definition
Let S = cfw_v1, v2, ., vn. Then S is a linearly dependent set if
and only if the vector equation c1v1+c2v2 + . + cnvn = 0 has a
solution with at least one ci 0.
Definiti
Basis of a vector space
A basis for a vector space, is a minimal set of vectors in V
that also spans V. That is, a basis for V is a set of vectors such
that the vectors are:
linearly independent;
span
Test #2 (
/30)
Name (print): _
Part A: Multiple Choice/Fill In The Blank (1 Mark Each)
Question 1:
Which of the following is a solution to:
1 0 2 7
3
1
2
0
3
1
[
]
a)
[1]
b)
[0]
c)
1 1 1 6
2
3
0
1 1 3
Test #3 (
/30)
Name (print): _
Student Number:
_
Part A: Multiple Choice/Fill In The Blank (1 Mark Each)
Question 1:
Consider two spans cfw_1 , , and cfw_1 , , . If [1 | 1 ] is
consistent for all , t
What is a Complex Number?
We denote the space of complex numbers with the bolded C, and it is simply the space
of all numbers written as c = a + bi (where i has the property: i2 = -1)
Here we call Re(
Polar Form
From the Cartesian plane, we can construct a new way of viewing complex numbers.
Standard form:
-2+2i
8
2
2
Rather than viewing a + bi, let us consider measuring the angle from the positive
11/01/2016
Matrix Notation
A matrix is also a storing device, but it has multiple columns. In this case, each
row represents one type of data, and each column represents another
type of data.
Coeffici
Chapter 8.1 Part 2 Q3 i: Determine which of the following are subspaces of a known vector space using the subspace test:
i)
| + =0
Problem Solving - What are the terms/strategies I may need? What
Chapter 8.1 Part 2 Q1 ii:
Consider a space with the following addition and scalar multiplication. Are the vector properties true of false?
+
=
=
c) There is a zero vector 0 such
Problem Solving - What are the terms/strategies I may need? What do I know?
Inequality Properties:
O1. Given a or b either a b or b a
O2. If a b and b a then a = b
O3. If a b and b c then a c
O4. If a
CARLETON UNIVERSITY
FINAL
EXAMINATION
SAMPLE
DURATION:
3 HOURS
No.
of Students
Department & Course Number:
Mathematics and Statistics MATH 1107
Course Instructor(s) Kevin Cheung
AUTHORIZED MEMORANDA
O
MATH 1107 - Fall 2016 Final Practice Problems
Answers and solution sketches are for reference only.
They are by no means models of proper solutions for obtaining full marks.
"
#
1 1
10
1. (a) Let A =.