MATH 1302 - PRACTICE THIRD MIDTERM EXAM WINTER 2008
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First Name
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Sample Instructions (instructions on the real exam will be similar to these): (1) All work to be considered for grading
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 t
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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
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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: le
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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 (s
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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
correspo
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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
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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 o
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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 equati
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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?
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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.
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 correspondi
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 gradin
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,
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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 citiz