Linear equality
an expression where all variables are to the first power And equality meaning equal sign ex, 2x+4=10
Linear inequality a linear expression with an inequity sign ex. 2x>10 Half plane the region defined by a linear inquality
Syste
Sec 4.2
Saturday, February 06, 2010
11:14 PM
Let v1, v2, . . ., vk be vectors in Rn. A vector v is called a linear
combination of v1, v2, . . ., vk if
v = c1 v1 + c2 v2 + . . . + ck vk
for some scalars (i.e. real numbers) c1, c2, . . ., ck
Section4dot2 Pa
Sec 4.2.REV
Monday, February 15, 2010
10:15 PM
(recall)
Let v1, v2, . . . , vk be some vectors in R n. A vector v in Rn is called a linear
combination of v1, v2, . . . , vk if
v = c1 v1 + c2 v2 + . . . + ck vk
for some real numbers (scalars) c1, c2, . . .
Sec 4.3
Saturday, February 06, 2010
11:23 PM
Let V be a vector space with operations
and
and and let W be a non-empty
set of vectors of V. If W is a vector space with respect to the operations
in V, then W is called a subspace of V.
Section4dot3 Page 1
an
Sec 4.4
Saturday, February 06, 2010
11:23 PM
Solution set: cfw_ ( -2 t + s + 1, t, s ) | t and s are in R
Section4dot4 Page 1
2 parameters a plane
4.4.1
Saturday, February 06, 2010
11:23 PM
Solution set: cfw_ ( -2 t + s + 1, t, s ) | t and s are in R
x
Sec 4.VS
Saturday, February 06, 2010
11:23 PM
A real vector space is a set V of objects together with two operations
and
satisfying the following properties: (for all u, v, and w in V and real numbers c and d)
V is closed under the operation
is commutativ
Sec 5.1
Saturday, February 06, 2010
11:13 PM
Let u = ( u1, u2, ., un ) and v = ( v1, v2, ., vn ) be vectors in Rn .
The dot product. u v is defined to be:
u v = u 1v 1 + u 2v 2 + . . . + u nv n
Section5dot1 Page 1
5.1.1
Saturday, February 06, 2010
11:13 P
Sec 5.2.p1
Saturday, February 06, 2010
11:23 PM
A unit vector is a vector of length 1
REMARK: If u is a unit vector, then | u | = 1
therefore | u | 2 = 1, and so u u = 1
All the standard basis vectors in R n are unit vectors
Section5dot2Part1 Page 1
5.2.p
Sec 5.2.p2
Sunday, February 07, 2010
12:03 AM
Let cfw_ v1 , v2 be an orthonormal set. If u is in spancfw_ v1 , v2 , then:
u = ( u v1 ) v1 + ( u v2 ) v2
Section5dot2Part2 Page 1
5.2.p2.1
Sunday, February 07, 2010
12:03 AM
Let cfw_ v1 , v2 be an ortho
Math 125 Fall 2014 Homework 8 Solutions
(1) An economy has three industries: electricity, oil, and plastic. Suppose that the production of $1 worth of electricity requires $.40 of electricity, $.50 of oil, and $.20 of
plastic. The production of $1 worth o
Math 125 Fall 2014 Homework 9 Solutions
(1) For each of the transition matrices below, compute the corresponding exact stable
vector:
.5 .2 .3
(a) T = .3 .8 .3
.2 0 .4
~ =
~
To find the stable
vector we solve T S
S.
.5 .2
.3
x
0
~ = ~0 .3 .2 .3 y = 0
Math 125 Fall 2014 Homework 6
Due Friday, October 15th, 11:55pm
(1) Given the 4-dimensional points P1 = (4, 2, 3, 1) and P2 = (3, 0, 2, 5) find the following:
(a) Find the vector ~v from P1 to P2 and compute its length.
P1 P2 = P2 P1 = (3, 0, 2, 5) (4, 2,
Math 125 Fall 2014 Homework 10 Solutions
Due Monday, December 7th, 11:59pm
(1) For the matrices below, let A = [a~1 a~2 . a~n ]. Use the given information to find a
solution to the system of equation A~x = ~b. Show all work and explain your reasoning.
-3
Math 125 Fall 2014 Homework 6
Due Friday, October 15th, 11:55pm
(1) Given the 4-dimensional points P1 = (4, 2, 3, 1) and P2 = (3, 0, 2, 5) find the following:
(a) Find the vector ~v from P1 to P2 and compute its length.
(b) Find 4 different vector equatio
Math 125 Fall 2014 Homework 7
Due Friday, October 31st, 11:55pm
(1) Find the indicated matrices below, or explain why they do not exist. Do not use your
calculator, and show all work.
3 2
3
5
5 1 6
2 5
7 C=
A=
B= 0
D = 0 E = 12
0 7 12
7 12
4 3
4
6
(a) E
Math 125 Fall 2014 Homework 10
Due Friday, December 5th, 11:55pm
(1) For the matrices below, let A = [a~1 a~2 . a~n ]. Use the given information to find a
solution to the system of equation A~x = ~b. Show all work and explain your reasoning.
-3
2 ~
-1
(a)
Sec 4.1
Saturday, February 06, 2010
11:10 PM
A vector is an ordered n-tuple of numbers ( u1, u2, ., un )
("pair" n = 2, or "triple" n = 3, or "quadruple", etc.)
Section4dot1 Page 1
4.1.1
Saturday, February 06, 2010
11:10 PM
A vector is an ordered n-tuple
Sec 3.6
Saturday, February 06, 2010
11:17 PM
Let cn = % of Coke drinkers at the end of month n
pn = % of Pepsi drinkers at the end of month n
The initial conditions or initial market distribution is:
c0 = .6 and p0 = .4
NOTE: the subscript 0 indicates "at
Probability Addition rule Multiplication rule Tree diagrams Binomial distribution Mutually exclusive Dependant Independent Most important 1. binomial distribution 2. tree diagrams 3. Multiplication rule 4. addition rule Least important The biggest ma
Sec 1.3
Wednesday, February 03, 2010
11:40 PM
Section1dot3 Page 1
1.3.1
Wednesday, February 03, 2010
11:40 PM
LINEAR - describes a mathematical expression in which all variables appear to the first power
EX: -2x + 5y = 12
Section1dot3 Page 2
1.3.2
Wednesd
Sec 1.4
Thursday, February 04, 2010
8:41 PM
Section1dot4 Page 1
1.4.1
Thursday, February 04, 2010
8:41 PM
A f unction, where every variable in the expression defining the function is
raised to the first power (at most)
EX: f(x, y) = 2x - (5/7)y
Section1do
Sec 1.5
Thursday, February 04, 2010
9:44 PM
1. Assign variables (detail and with units)
2. Define the objective function (what one is trying to optimize)
3. Define the constraints (the system of inequalities)
4. Solve the linear program
Section1dot5 Page
Sec 2.2
Friday, February 05, 2010
4:32 PM
A matrix is in reduced echelon form if it is in row echelon form and has the
additional property that above each leading 1 are 0's
This is also called reduced row echelon form (rref)
Section2dot2 Page 1
2.2.1
Frid
Sec 2.3
Tuesday, February 02, 2010
7:45 PM
A system of linear equations is consistent if it has one or more solutions
A linear system that does not have any solutions is called inconsistent
Section2dot3 Page 1
2.3.1
Tuesday, February 02, 2010
7:45 PM
A sy
Sec 2.4
Friday, January 29, 2010
11:01 PM
Section2dot4 Page 1
2.4.1
Wednesday, February 03, 2010
3:12 PM
The first octant is the portion of
R3 defined by x 0, y 0, and z 0
Section2dot4 Page 2
2.4.2
Wednesday, February 03, 2010
3:12 PM
The first quadrant i
Sec 2.5
Friday, February 05, 2010
5:02 PM
Pivoting is the procedure of choosing a point in a matrix, turning it into a 1,
and then annihilating the entries above and below it.
The chosen point / entry is referred to as "the pivot point / entry "
or just "
Sec 2.6
Friday, February 05, 2010
5:17 PM
An algebraic method for finding a maximum in a linear program
with two or more variables
Now we will learn a scaled down version. The full version is in Chapter 4
Section2dot6 Page 1
2.6.1
Friday, February 05, 201