3. Linear Transformations
So far we have seen a few linear transformations, but what makes them LINEAR?
To be linear, a transformation must have the following properties:
T(u+v)= T(u)+T(v) For any vectors u and v in the domain of T
T(cu)=cT(u) For all scalars c and every vector u in the domain of T
The basic idea is that for vector addition and scalar multiplication, the results are the same
if you perform the operation before or after you apply the transformation.
This gives an easy way to test a transformation for linearity. It is always the case that
T(0)=0
Also, a lineartransformation alwaysmaps lines to lines (or to zero)
4. Linear Transformations
DEFINITIONS:
A mapping T:Rn
↦Rm
is said to be ONTO if each b in Rm
is the image
of at least one x in Rn
.
Domain
ℝn Range is
All of ℝm
T
Domain
ℝn
Range is a
subspace of ℝm
T
T is onto T is not onto
5. Linear Transformations
A mapping T:Rn
↦Rm
is said to be ONE-TO-ONE if each b in Rm
is the
image of at most one x in Rn
.
Tisone-to-one
T
T is not one-to-one
T
6. Linear Transformations
A couple of quick tests to see if a transformation is one-to-one or
onto:
More Columns than Rows – it can’t be One-to-One
More Rows than Columns – it can’t be Onto More precisely:
A transformation is onto if the columns of A span Rm
.
A transformation is one-to-one if the columns are linearly independent.
7. Matrix Transformations
For each x in Rn , T(x) is computed as Ax, where A is an m*n
matrix.
RFor simplicity, we denote such a matrix transformation by
x↦Ax.
The domain of T is Rn when A has n columns and the codomain
of T is Rm when each column of A has m entries.
So an m*n matrix transforms vectors from Rn into vectors from
Rm.
8.
9. Application of Matrices
Cryptographyis the process of encryptingdataso that third party
can’t read it and privacy can be maintained.
It was started with the TV cable industrieswhere even people
who were not the customer could watch the TV programs
So, Videocipher encryptionsystemwas invented which would
convert signals intodigital form i.e. encrypt it, and the data were
send over the satellite. The Videocipher box would decrypt the
signal and those satellite dish owner whohad Videocipher box
would receive the decrypted signali.e. the originalsignalbefore
encryption.
In matrix same thing can be done by this Process, But
there are many other methodsfor cryptography
10. Computer Animations
Matrix transforms are very useful within the world of
computer graphics. Software and hardware graphics
processor uses matrices for performing operations
such as scaling, translation, reflection and rotation.
11. Since a digital image is basically a matrix to begin with: The rows
and columns of the matrix correspond torows and columns of
pixels, and the numerical entriescorrespond tothe pixels’ color
values. Decoding digitalvideo, for instance, requiresmatrix
multiplication.
In the same way that matrix multiplication can help process digital
video, it can help process digitalsound. A digital audio signal is
basically a sequence of numbers, representing the variation over
time of the air pressure of an acoustic audio signal. Many
techniquesfor filtering or compressing digitalaudio signals, such as
the Fourier transform, rely on matrix multiplication.
13. Electronics
The behavior of many electronic components can be
described using matrices. Let A be a 2-dimensional vector
with the component's input voltage v1 and input current
i1 as its elements, and let B be a 2-dimensional vector with
the component's output voltage v2 and output current i2
as its elements.
Then its behaviour can be described by B = H · A,
where H is a 2 x 2 matrix containing one impedance
element (h12), one admittance element (h21) and two
dimensionless elements (h11 and h22). Calculating a
circuit now reduces to multiplying matrices.
15. Other uses…
Matrices are very useful for organization, like for
scientists who have to record the data from their
experiments if it includes numbers.
Stochastic matrices and Eigen vector solvers are
used in the page rank algorithms which are used
in the ranking of web pages in Google search.
And In architecture, matrices are used with
computing. If needed, it will be very easy to add
the data together, like with matrices in
mathematics.
