Appl pplica icatio tions ns of line linear ar algebra

Sangat Vaidya atmiya institute of technology and science Rajkot (Guj.)

11 july 2012 Applications

of

linear

algebra

What is linear algebra?

Linear algebra is a branch of mathematics charged with properties of ﬁnite dimensional vector spaces and linear mappings between such spaces.

Applications

of

linear

algebra

Theo Theory ry and and me meth thod odss of this this bran branch ch ar aree exten tensiv sively ely usef seful in engineering, natural sciences, computer sciences and the social social sciences sciences (particular (particularly ly economics) economics)..

Applications

of

linear

algebra

BASIC CONCEPTS OF DISTANCE Applications

of

linear

algebra

Distance is a numerical description of how far apart two objects are from each other. Distance provides a concrete way of describing what it means for two elements of some space to be ”close to” or ”far away from” each other.

Applications

of

linear

algebra

The distance between two points x(x1,y1,z1) and y(x2,y2,z2) is given by vector xy and is deﬁned by d ( x y) ,

Applications

=

( x1 − x2 )2 + ( y1 − y2 )2 + ( z1 − z2)2

of

linear

algebra

APPLICATIONS

Applications

of

linear

algebra

GLOBAL POSITIONING SYSTEM Applications

of

linear

algebra

what is GPS?

The Global Positioning System (GPS) is a spacebased satellite navigation system which provides location and time information of objects in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites.

Applications

of

linear

algebra

Applications

of

linear

algebra

The GPS system is maintained by the United States Government and is freely accessible by anyone through a GPS receiver.

Applications

of

linear

algebra

ARRANGEMENT OF GPS SATELITES SURROUNDING THE EARTHS GLOBE

Applications

of

linear

algebra

Applications

of

linear

algebra

HOW THE CONCEPT OF DISTANCE IS USED IN GPS ? Applications

of

linear

algebra

Let a ship is at an unknown point (x, y, z) at some time t.

Applications

of

linear

algebra

Let a ship is at an unknown point (x, y, z) at some time t.

Applications

of

linear

algebra

The GPS identiﬁes the ship’s coordinates (x, y, z) at any time t using a triangulation system and computed distances from four satellites.

Applications

of

linear

algebra

Applications

of

linear

algebra

These distances are computed using the speed of light (approximately 0.469 times Earth radius per hundredth of a second) and the time it takes for the signal to travel from the satellite to the ship.

Applications

of

linear

algebra

If the the signal transmitted by the satellite at time t0 is received by the ship at time t, then the distance d travelled by the signal will be d = 0 469(t − t 0) .

Applications

of

linear

algebra

Applications

of

linear

algebra

In theo theory ry,, knowi knowing ng thre threee shipship-to to-s -sat atel elli lite te dist distan ance cess would suﬃce to determine the three unknown coordinates of the ship.

Applications

of

linear

algebra

Applications

of

linear

algebra

However, the problem is that the ships (or other GPS users) do not generally have clocks that can compute t with suﬃcient accuracy for global positi sition onin ing. g. Thus Thus,, the vari variab able le t must must be regar regarde ded d as a fourth unknown, and hence the need for the distance to a fourth satellite.

Applications

of

linear

algebra

Applications

of

linear

algebra

The distance from one satellite can be computed as d ( x y) = ( x − x0 )2 + ( y − y0)2 + ( z − z0)2 ,

Applications

of

linear

algebra

If we now equate the squares of d from both equations and round oﬀ to three decimal places, then we obtain

Applications

of

linear

algebra

If we now equate the squares of d from both equations and round oﬀ to three decimal places, then we obtain ( x − x0 )2 + ( y − y0 )2 + ( z − z0)2

Applications

of

linear

=

0 22(t − t 0 )2 .

algebra

Applications

of

linear

algebra

Since there are four diﬀ erent erent satellites, and we can get an equation like this for each one, we can produce four equations in the unknowns x, y, z and t0 which produces a system of four linear equations in four unknowns x ,y, z and t0 that can be solved by algebraic or numerical methods.

Applications

of

linear

algebra

Applications

of

linear

algebra

Applications

BASIC CONCEPT OF LINEAR COMBINATION OF VECTORS of

linear

algebra

Applications

of

linear

algebra

A vector w is called a linear combination of the vectors v1,v 2,,vr if it can be expressed in the form w = k 1 v1 + k 2v2 + + k r r vr where k1,k2,..,kr are scalars. ...

Applications

of

linear

algebra

ILLUSTRATION

Applications

of

linear

algebra

ILLUSTRATION

If i=(1,0,0), j=(0,1,0), k =(0,0,1) then v=(a,b,c) can be expressed as linear combination of i, j and k since v

Applications

=

(a b c) ,

,

=

of

a(1 0 0) + b(1 0 0) + c(1 0 0) = ai + bj + ck ,

,

,

linear

,

,

algebra

,

Applications

of

linear

algebra

APPLICATIONS

Applications

of

linear

algebra

Applications

of

linear

algebra

RGB COLOR MODEL

Applications

of

linear

algebra

WHAT IS RGB COLOR MODEL ?

Applications

of

linear

algebra

WHAT IS RGB COLOR MODEL ?

The RGB color model is an additive color model in which red , green and blue colors are added together in various combi mbinations to produce a broad arra ar ray y of color colors. s. The The name name of the the mode modell come comess from the initials of the three additive primary colors RED, GREEN, and BLUE.

Applications

of

linear

algebra

Applications

of

linear

algebra

Applications

of

linear

algebra

Applications

of

linear

algebra

The main function of the RGB color model is in the sensing, representation, and display of images in electronic systems such as televisions and computers.

Applications

of

linear

algebra

Applications

of

linear

algebra

HOW IS LINEAR COMBINATION OF VECTORS USED IN RGB MODEL ? Applications

of

linear

algebra

Applications

of

linear

algebra

Colors in this system are created by adding together diﬀ erent erent percentages of the primary colors RED (R), GREEN (G), and BLUE (B). One way to do this is to identify the primary colors with the vectors r = (1 0 0) (pure red) g = (0 1 0) (pure green) b = (0 0 1) (pure blue) ,

Applications

of

,

,

,

,

,

linear

algebra

Applications

of

linear

algebra

To create all other colors by forming linear combinations of r, g, and b using coe ﬃcients between 0 and 1, inclusive; these coeﬃcients represent the percentage of each pure color in the mix.

Applications

of

linear

algebra

Applications

of

linear

algebra

The set of all such color vectors is called RGB space or the RGB color cube where each vector is represented as c = c1r + c2 g + c3b = (c1 c2 c3 ) 0≤ci ≤ 1 ,

Applications

of

,

linear

algebra

CMYK COLOR MODEL Applications

of

linear

algebra

Applications

of

linear

algebra

Applications

of

linear

algebra

Applications

of

linear

algebra

Color magazines and books are printed using CMY color model or FOUR color model.

Applications

of

linear

algebra

Color magazines and books are printed using CMY color model or FOUR color model. Colors in this model are created using four colored ored inks: inks: CYAN( CYAN(C), C), MAGEN MAGENT TA(M), A(M), YELLO YELLOW( W( and BLACK(K).

Applications

of

linear

algebra

Applications

of

linear

algebra

The colors can be created by printing dot patterns (called rosettes) with the four colors and allowing the reader’s eye and perception process to create the desired color combination.

Applications

of

linear

algebra

Applications

of

linear

algebra

The color is decided by associating the 4 base colors with vectors c = (1 0 0 0) (cyan) m = (0 1 0 0) (magenta) y = (0 0 1 0) (yellow) k = (0 0 0 1) (black) and describing the ink color as a linear combination of these using coeﬃcients between 0 and 1, inclusive. ,

,

,

of

,

,

,

Applications

,

,

,

,

,

,

linear

algebra

Applications

of

linear

algebra

Thus, an ink color p is represented as a linear combination of the form p = c1 c + c2m + c3 y + c4 k where 0 ≤ ci ≤ 1 The set of all such linear combinations is called CMYK space.

Applications

of

linear

algebra

Applications

of

linear

algebra

DIFFERENCES BETWEEN RGB AND CMYK COLOR MODELS Applications

of

linear

algebra

Applications

of

linear

algebra

RGB RGB is ADDITIVE color model. RGB mostly used to display graphics. graphics.

Applications

of

linear

algebra

CMYK RGB RGB is ADDITIVE color model. RGB mostly used to display graphics. graphics.

Applications

of

CMYK is TRACTIVE model.

SUBcolor

CMYK is mostly used for printing purpose.

linear

algebra

Applications

of

linear

algebra

RGB It starts with the absence of light or color - black - as one would ﬁnd in a television screen or computer monitor with no power, and adds light in colors as needed.

Applications

of

linear

algebra

CMYK RGB It starts with the absence of light or color - black - as one would ﬁnd in a television screen or computer monitor with no power, and adds light in colors as needed.

Applications

of

It starts with white such as paper - which is the combination of all all colo colors rs,, and and ﬁlte ﬁlters rs of subtracts out those colors not wanted by the application of colored dyes or inks. linear

algebra

REFERENCES

Applications

of

linear

algebra

REFERENCES

H. Anton and C. Rorres , Elementary Linear Algebra (Applications version),Wiley India, (2009).

Applications

of

linear

algebra

REFERENCES

H. Anton and C. Rorres , Elementary Linear Algebra (Applications version),Wiley India, (2009). G. Strang, Linear algebra and its applica icatio tions (4th Edition), Thomson(2006).

Applications

of

linear

algebra

REFERENCES

H. Anton and C. Rorres , Elementary Linear Algebra (Applications version),Wiley India, (2009). G. Strang, Linear algebra and its applica icatio tions (4th Edition), Thomson(2006). S. Kumaresan, Linear algebra - A Geometric approach, Prentice Hall of India (2000).

Applications

of

linear

algebra

Applications

of

linear

algebra

THANK YOU

Applications

of

linear

algebra