Appl pplica icatio tions ns of line linear ar algebra
Sangat Vaidya atmiya institute of technology and science Rajkot (Guj.)
11 july 2012 Applications
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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.
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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)..
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BASIC CONCEPTS OF DISTANCE Applications
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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.
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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) ,
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( x1 − x2 )2 + ( y1 − y2 )2 + ( z1 − z2)2
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APPLICATIONS
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GLOBAL POSITIONING SYSTEM Applications
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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.
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The GPS system is maintained by the United States Government and is freely accessible by anyone through a GPS receiver.
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ARRANGEMENT OF GPS SATELITES SURROUNDING THE EARTHS GLOBE
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HOW THE CONCEPT OF DISTANCE IS USED IN GPS ? Applications
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Let a ship is at an unknown point (x, y, z) at some time t.
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Let a ship is at an unknown point (x, y, z) at some time t.
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The GPS identifies the ship’s coordinates (x, y, z) at any time t using a triangulation system and computed distances from four satellites.
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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.
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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) .
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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.
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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.
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The distance from one satellite can be computed as d ( x y) = ( x − x0 )2 + ( y − y0)2 + ( z − z0)2 ,
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If we now equate the squares of d from both equations and round off to three decimal places, then we obtain
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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
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0 22(t − t 0 )2 .
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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.
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BASIC CONCEPT OF LINEAR COMBINATION OF VECTORS of
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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. ...
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ILLUSTRATION
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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
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(a b c) ,
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a(1 0 0) + b(1 0 0) + c(1 0 0) = ai + bj + ck ,
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APPLICATIONS
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RGB COLOR MODEL
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WHAT IS RGB COLOR MODEL ?
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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.
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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.
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HOW IS LINEAR COMBINATION OF VECTORS USED IN RGB MODEL ? Applications
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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) ,
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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.
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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 ,
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CMYK COLOR MODEL Applications
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Color magazines and books are printed using CMY color model or FOUR color model.
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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).
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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.
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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. ,
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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.
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DIFFERENCES BETWEEN RGB AND CMYK COLOR MODELS Applications
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RGB RGB is ADDITIVE color model. RGB mostly used to display graphics. graphics.
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CMYK RGB RGB is ADDITIVE color model. RGB mostly used to display graphics. graphics.
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CMYK is TRACTIVE model.
SUBcolor
CMYK is mostly used for printing purpose.
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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.
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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.
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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
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REFERENCES
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REFERENCES
H. Anton and C. Rorres , Elementary Linear Algebra (Applications version),Wiley India, (2009).
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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).
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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).
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THANK YOU
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