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.
If
, then
contains a basis.
Questions & Answers
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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