apxD copy - NUMERICAL MATHEMATICS & COMPUTING 6th...

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NUMERICAL MATHEMATICS & COMPUTING 6th Edition Ward Cheney/David Kincaid c ± UT Austin Engage Learning: Thomson-Brooks/Cole www.engage.com www.ma.utexas.edu/CNA/NMC6 September 1, 2011 Ward Cheney/David Kincaid c ± (UT Austin[10pt] Engage Learning: Thomson-Brooks/Cole www.engage.com[10pt] www.ma.utexas.edu NUMERICAL MATHEMATICS & COMPUTING 6th Edition September 1, 2011 1 / 48
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Appendix D. Linear Algebra Concepts and Notation The two concepts from linear algebra that we are most concerned with are vectors matrices Their usefulness is in compressing complicated expressions into a compact notation. Here the vectors and matrices are most often real consisting of real numbers. These concepts easily generalize to complex vectors and matrices. Ward Cheney/David Kincaid c ± (UT Austin[10pt] Engage Learning: Thomson-Brooks/Cole www.engage.com[10pt] www.ma.utexas.edu NUMERICAL MATHEMATICS & COMPUTING 6th Edition September 1, 2011 2 / 48
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Vectors A vector x 2 R n can be thought of as a one-dimensional array of numbers written as x = 2 6 6 6 4 x 1 x 2 . . . x n 3 7 7 7 5 where x i is called the i th element , entry ,o r component . An alternative notation that is useful in pseudocodes is x =( x i ) n Sometimes the vector x displayed above is said to be a column vector to distinguish it from a row vector y written as y =[ y 1 , y 2 ,..., y n ] Ward Cheney/David Kincaid c ± (UT Austin[10pt] Engage Learning: Thomson-Brooks/Cole www.engage.com[10pt] www.ma.utexas.edu NUMERICAL MATHEMATICS & COMPUTING 6th Edition September 1, 2011 3 / 48
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For example, here are some vectors: 2 6 6 6 6 6 4 1 5 3 - 5 6 2 7 3 7 7 7 7 7 5 , [ , e , 5 , - 4] , " 1 2 1 3 # Ward Cheney/David Kincaid c ± (UT Austin[10pt] Engage Learning: Thomson-Brooks/Cole www.engage.com[10pt] www.ma.utexas.edu NUMERICAL MATHEMATICS & COMPUTING 6th Edition September 1, 2011 4 / 48
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To save space, a column vector x can be written as a row vector such as x =[ x 1 , x 2 ,..., x n ] T or x T x 1 , x 2 x n ] By adding a T (for transpose ) to indicate that we are interchanging or transposing a row or column vector. As an example, we have [1 2 3 4] T = 2 6 6 4 1 2 3 4 3 7 7 5 Ward Cheney/David Kincaid c ± (UT Austin[10pt] Engage Learning: Thomson-Brooks/Cole www.engage.com[10pt] www.ma.utexas.edu NUMERICAL MATHEMATICS & COMPUTING 6th Edition September 1, 2011 5 / 48
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Many operations involving vectors are component-by-component operations . For vectors x and y x = 2 6 6 6 4 x 1 x 2 . . . x n 3 7 7 7 5 , y = 2 6 6 6 4 y 1 y 2 . . . y n 3 7 7 7 5 the following de±nitions apply. Ward Cheney/David Kincaid c ± (UT Austin[10pt] Engage Learning: Thomson-Brooks/Cole www.engage.com[10pt] www.ma.utexas.edu NUMERICAL MATHEMATICS & COMPUTING 6th Edition September 1, 2011 6 / 48
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Equality x = y if and only if x i = y i for all i Inequality x < y if and only if x i < y i for all i Addition/Subtraction x ± y = 2 6 6 6 4 x 1 ± y 1 x 2 ± y 2 .
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apxD copy - NUMERICAL MATHEMATICS &amp; COMPUTING 6th...

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