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Notes_05_27

Notes_05_27 - U Washington AMATH 352 Spring 2009 Applied...

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U. Washington AMATH 352 - Spring 2009 Applied Linear Algebra and Numerical Analysis AMATH 352 - Spring 2009 Prof. U. Hetmaniuk University of Washington The course goals are to understand the basic concepts of linear algebra and to obtain an introduction to some aspects of computational techniques used in matrix methods. We will study basic concepts in linear algebra, including vec- tors, vector spaces, linear transformations, matrix-vector manipulations, solving linear systems, least squares problems, and eigenvalue problems. Matrix decom- positions (e.g. LU, QR, SVD, etc.) will play a fundamental role throughout the course. The emphasis will be on practical aspects of linear algebra and nu- merical methods for solving these problems. Such problems arise constantly in science, engineering, finance, computer graphics. Professor Randall J. LeVeque (U. Washington) created the notes. He, kindly, allowed me to modify them. All the mistakes and typos are mine. 1

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U. Washington AMATH 352 - Spring 2009 1 Column Vectors In geometry, the two-dimensional plane is denoted R 2 and the three-dimensional space R 3 . A vector is an object comprised of a magnitude and a direction. In the plane, we can draw a vector as an arrow with some length and pointing somewhere . A vector can also be thought of as a displacement. A displacement does not depend where it starts. Consequently, the vectors are equal, even though they start from different places, because they have equal length and equal direction. The basic idea here, combining magnitude with direction, is the key to extending to higher dimensions. In this section, we define the generalization of vectors in the two-dimensional plane and three-dimensional space. Let m be a positive integer. We denote R m the set of all real m -tuples, i.e. the set of all sequences with m components, each of which is a real number. The standard notation for an element x of R m is the column vector notation: x R m ⇐⇒ x = x 1 . . . x m . (1) It is important to remember that, in many applications of linear algebra, the elements of the vector represent something different from the three physical co- ordinates of ordinary space. There is often nothing unphysical about considering vectors with many more than 3 components. Example. We have - 1 3 R 2 , 7 0 3 R 3 , and 1 - 2 / 5 - 3 / 5 4 R 4 . 2
U. Washington AMATH 352 - Spring 2009 1.1 Addition of column vectors We can define the addition of two vectors x and y of R m : x = x 1 . . . x m y = y 1 . . . y m , x + y = x 1 + y 1 . . . x m + y m . (2) The set R m is closed under addition, meaning that whenever the addition is applied to vectors in R m , we obtain another vector in the same set R m . For example, we have 1 2 3 + 2 - 4 8 = 1 + 2 2 - 4 3 + 8 = 3 - 2 11 .

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Notes_05_27 - U Washington AMATH 352 Spring 2009 Applied...

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