Vector spaces are the principal object of study
in linear algebra. A vector space is always defined with respectto a field of scalars.
Fields
A field is a set
equipped with two operations, addition and
mulitplication, and containing two special members 0 and 1(
), such that for all
there exists
such that
there exists
such that
More concisely
is an
abelian group under addition
is an
abelian group under multiplication
multiplication distributes over addition
Examples
,,
Vector spaces
Let
be
a field, and
a
set. We say
is a vector space over
if there exist two operations, defined for all
,
and
:
vector addition: (
,
)
scalar multiplication:
(
,
)
and if there exists an element denoted
, such that the following hold for all
,
, and
,
, and
there exists
such that
More concisely,
is an abelian
group under plus
Natural properties of scalar multiplication
Examples
is a vector space over
is a vector space over
is a vector space over
is
not a vector space
over
The elements of
are called
vectors .
Euclidean space
Throughout this course we will think of a signal
as a vector
The samples
could be samples from a finite duration, continuous
time signal, for example.
A signal will belong to one of two vector spaces:
Real euclidean space
(over)
Complex euclidean space
(over)
Subspaces
Let
be a vector
space over
.
A subset
is called a
subspace of
if
is a vector space over
in its own right.
,
,
.
is any line
through the origin.
Are there other subspaces?
is a subspace if and only if for all
and
and for all
and
,
Linear independence
Let
.
We say that these vectors are
linearly
dependent if there exist scalars
such that
and at least one
.
If
only holds for the case
, we say that the vectors are
linearly
independent .
so these vectors are linearly dependent in
.
Spanning sets
Consider the subset
. Define the
span of
Fact:
is a subspace of
.
,
,
,
,
.
is the xy-plane.
Aside
If
is infinite, the notions of
linear independence and span are easily generalized:
We say
is linearly independent if, for
every finite collection
, (
arbitrary) we
have
The span of
is
In both definitions, we only consider
finite sums.
Bases
A set
is called a
basis for
over
if and only if
is linearly independent
Bases are of fundamental importance in signal processing. They
allow us to decompose a signal into building blocks (basisvectors) that are often more easily understood.
= (real or complex) Euclidean
space,
or
.
where the 1 is in the
position.
over.
which is the DFT basis.
where
.
Key fact
If
is a basis for
,
then every
can be written uniquely (up to order of terms) in
the form
where
and
.
Other facts
If
is a
linearly independent set, then
can be extended to a basis.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .