MATH 1302 - PRACTICE THIRD MIDTERM EXAM WINTER 2008
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Sample Instructions (instructions on the real exam will be similar to these): (1) All work to be considered for grading must be written in the space provided. The reverse sid
MATH 1302 - PRACTICE THIRD MIDTERM EXAM SOLUTIONS WINTER 2008
1. Suppose an economy consists of three sectors manufacturing, agriculture and services with unit consumptions as given in the following table. Purchased Inputs Consumed per Unit of Output From
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Reduced row echelon form
Theorem:
o Every matrix is row equivalent to exactly one
reduced echelon matrix.
o Note: a matrix may be row equivalent to many
echelon matrices
o Ex.
o All are in echelon form, but only the far leftmost
matrix i
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Matrix Terminology
Zero row or column: row or column with all entries
equal to zero.
Nonzero row or column: row or column with at least
one nonzero entry
Leading entry of a row: leftmost nonzero entry.
Terminology
Row Echelon form
o A
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Logic
If P and Q are statements, then P = Q means that
statement P implies statement Q.
EX. xZ = xR
o since the statement x is an integer implies the
statement x is a real number (since all integers
are real numbers)
P Q means that sta
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Row Equivalence and Solution sets
If a matrix A is obtained from a matrix B by row
operations, then any solution of the system
corresponding to A is a solution of the system
corresponding to B.
Since A can also be obtained from B by row
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Solving Linear Systems
Recall the algorithm for solving linear systems:
Important:
Our algorithm is precise and involves no guesswork. Its just
a series of steps to follow (compare to differentiation versus
integration)
Even more importan
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Different Algorithms
Our algorithm allows you to reduce any augmented
matrix to EF or RREF.
The algorithm is not unique there are other ones
(e.g. algorithm described in the proof of Theorem
REMEF in the FCLA text).
Since the RREF of a
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Linear Algebra What? Discrepancies between
Algebra
Branch of mathematics studying structure, relations
and quantity
Symbolic operations and variables
Systematic solutions of equations and algorithms.
Linear
Geometric; which relates to
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Linear Equations
What is a linear Equation
Should be the equation of a linear geometric object
(i.e. A line, plane etc.)
How do we know the equation just by looking at an
equation?
How can we make this purely algebraic definition
more
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One Element Sets
Sets can have a single element.
EX. cfw_4 is the set that contains only the number 4
There is a difference between a single element set and
the element itself
EX. The set cfw_4 is not the same as the number 4 itself
MATH 1302 - PRACTICE FIRST MIDTERM EXAM SOLUTIONS WINTER 2008 1. a) If a linear system consists of 4 equations and 5 unknowns, what is the maximum possible number of pivot positions in the corresponding coecient matrix? b) If a linear system consists of 1
MATH 1302 - PRACTICE SECOND MIDTERM EXAM WINTER 2008
Surname
First Name
Student #
Sample Instructions (instructions on the real exam will be similar to these): (1) All work to be considered for grading must be written in the space provided. The reverse si
MATH 1302 - PRACTICE SECOND MIDTERM EXAM SOLUTIONS WINTER 2008 1. Let A= 3 7 1 , 0 4 8 1 2 1/2 0 , B = 8 0 3 2/3 5 20 C = 1 1 , 34 D= 2 8 4 03 4
a) Which of the following operations are dened? AB, BA, AC, CA, AD, DA For those that are dened, what is the s
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Basic Set Theory
A set is a collection of objects. These objects are called the
elements of the set and can be mathematical, physical, or
anything else.
The set of all Canadian citizens. Its members are the
people who are Canadian citize