Mathematics Course 111: Algebra I
Part IV: Vector Spaces
D. R. Wilkins
Academic Year 1996-7
9
Vector Spaces
A vector space over some field K is an algebraic structure consisting of a set V on which are defined
two algebraic operations: a binary operation
Accurate Electronic, transport, and Related Properties of Wurtzite Beryllium Oxide
(w-BeO)
Cheick Oumar Bambal, fuchard Inakpenul, Yacouba I. Diakite2, Yr'[iy Malozovsky', Lashouda
Franklin', Diola Bagayoko'.
lDepartment of Mathematics and Physics, Southe
Section 2.2 Multiplication Property of Equality
Objective #1: Use the Multiplication Property of Equality
A. Use multiplication property to solve equations.
If a=b , then a c=b c where a, b, and c are real numbers.
The multiplication property is used to
Section 2.1 ADDITION PROPERTY OF EQUALITY
Objective #1: ADDITION PROPERTY
A. A linear equation in one variable is a equation that can be written in
the form ax +b=c where a, b, and c are real numbers and a 0.
Equations
3x1=17
Expressio
ns
3x13
8+16=16+8 y
1.7 Properties or Real Number
Objective #1: Commutative and Associative Property
Commutative property
Addition: a + b = b + a
Multiplication: a b = b a
Associative property
Addition: (a + b) + c = a + (b + c)
Multiplication: (a b) c = a (b c)
Examples:
Section2.3FurtherSolveLinearEquations
Objective #1: Apply the General Strategy for Solving a Linear Equation
General Procedure:
1. If the equation contains a fraction multiply both sides of the equation by the LCD to
_ the fraction in the equation. If the
EE210: Switching Systems
Lecture 11: Decoders
Prof. YingLi Tian
Mar. 22, 2016
Department of Electrical Engineering
The City College of New York
The City University of New York (CUNY)
1
1-bit Adder
One-bit adder Truth Table
2
Delay in Combinational Logic C
Characteristics of a Successful Student
Many students in high school do not know what it takes to be successful in the school
environment. They understand good and bad grades in a general way, and they sense that they
should attend classes, but that is wh
Biology Review
Cell Structure and Function Review
1. Cell specialization means that cells in an
organism are uniquely suited to
a)carry on reproduction.
b)respond to changing conditions.
c)react with the environment.
d)perform a specific function.
Cell St
Managerial Accounting
Exam 4 Study Guide
Chapter 11
Know and understand what are the three factory overhead allocation methods
Know and understand how to calculate the overhead cost per unit for two different
products and know how to calculate the overhea
Review for Exam 3 Solutions, Spring 2014
Note: The problems on this review may or may not be similar to the problems on Exam 3.
1. The graph of a sine function is shown. The function is defined for < t < , but only
a partial graph is shown.
(a) What is th
Practice Quiz 3 Math 2312
Let the quadratic function f(x) = x2 3x be given.
a. Find the standard form for f(x). That is, find constants a, h, and k such that
f(x) = a(x h) 2 + k.
b. Find the zeros of f(x).
c. Sketch the graph of f(x)
using the axes below.
Practice Quiz 4 Math 2312
Let the following rational functions be given.
x +1
x2 +1
x3 1
x2 3
x2 2
x 1
f(x) =
, g(x) =
, h(x) =
, r(x) = 2
, s(x) = 2
, t(x) = 2
x 1
x 1
x +1
x +1
x 1
x 2
For each of the following conditions, list all the functions shown a
Review for Exam 2, Spring 2014
Note: The problems on this review may or may not be similar to the problems on
Exam 2.
2x 4
.
1. Let f(x) = x + 2
(a) What are the x- and y-intercepts of the function?.
(b) What is the equation of the vertical asymptote?
(c)
Review for Exam 2, Solutions, Spring 2014
Note: The problems on this review may or may not be similar to the problems on
Exam 2.
2x 4
.
1. Let f(x) = x + 2
(a) What are the x- and y-intercepts of the function?.
x-int: 2x 4 = 0 => x = 2, y-int: f(0) = 4/2
Review for Exam 1, Spring, 2013
Note: The problems on this review may or may not be similar to the problems on
Exam 1.
1. Use the graph of f(x) to the right to find
or estimate:
(a) f (4)
1
(b) f (4)
(c) (f f )(8) = f (f(8)
(d) the average rate of change
Review for Exam 1, Spring, 2013
Note: The problems on this review may or may not be similar to the problems on
Exam 1.
1. Use the graph of f(x) to the right to find
or estimate:
(a) f (4) = 24
1
(b) f (4) 7.5
(c) (f f )(8) = f (f(8) = f (0) = 32
(d) the a
Practice Quiz Math 2312 Solutions
Given the function defined by h(t) = t2 2t, evaluate the function at each specified value
of the independent variable. Show your work if you want partial credit.
a. h(2) = 22 2(2) = 4 4 = 0
b. h(t 1) = at2 + bt + c. Find
Review for Exam 3, Spring 2014
Note: The problems on this review may or may not be similar to the problems on Exam 3.
1. The graph of a sine function is shown. The function is defined for < t < , but only
a partial graph is shown.
(a) What is the period f
Name:_
Practice Quiz 6 Math 2312, Spring 2014
Fill in the blanks in the table. For each numbered row of the table, draw an
appropriate right triangle in the space provided. Your triangle need not be drawn to
scale. ( 0 90 , 0 2 )
Function
(deg)
(rad)
1.
Name:_
Practice Quiz 6, Math 2312, Spring 2013
Fill in the blanks in the table. For each numbered row of the table, draw an
appropriate right triangle in the space provided. Your triangle need not be drawn to
scale. ( 0 90 , 0 2 )
Function
(deg)
(rad)
F
Practice Quiz 5 Solutions
Show your work for partial credit.
In Problems 1, 2, and 3, use the properties of logarithms to expand the expression
as a sum, difference and/or constant multiple of logarithms. (Assume all variables
are positive.)
3
1. log 5x y
Practice Quiz 5
Show your work for partial credit.
In Problems 1, 2, and 3, use the properties of logarithms to expand the expression
as a sum, difference and/or constant multiple of logarithms. (Assume all variables
are positive.)
3
1. log 5x y
2.
3.
log
Practice Quiz Math 2312
Given the function defined by h(t) = t2 2t, evaluate the function at each specified value
of the independent variable. Show your work if you want partial credit.
a. h(2)
b. h(t 1) = at2 + bt + c. Find the values of the constants a,
Practice Quiz 4 Math 2312
Let the following rational functions be given.
x +1
x 2 +1
x 3 1
x2 3
x2 2
x 1
f(x) =
, g(x) =
, h(x) =
, r(x) = 2
, s(x) = 2
, t(x) = 2
x 1
x 1
x +1
x +1
x 1
x 2
For each of the following conditions, list all the functions shown