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Slides_2010_01_06

# Slides_2010_01_06 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis January 06, 2010 - Session 2 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 6, 2010 Comments Scorelator Accounts Displacement A vector can also be thought of as a displacement. How do we measure the length of a displacement? Column Vectors A vector is an object comprised of a magnitude and a direction. These vectors have equal length and direction. Combining magnitude with direction is the key to extending to higher dimensions. Column Vectors 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) Example We have- 1 3 R 2 , 7 3 R 3 , and 1- 2 / 5- 3 / 5 4 R 4 . Equality of Column Vectors We say that two vectors x and y of R m are equal if they satisfy x = x 1 . . . x m y = y 1 . . . y m , x = y iff x i = y i 1 i m . (2) Example x = iff x i = , 1 i m . 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 . (3) 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 ....
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Slides_2010_01_06 - Applied linear algebra and numerical...

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