LA07_1 - _ Linear Algebra: Spring 2007 Chapter 1. Vectors...

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___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 1-1 Chapter 1. Vectors 1. Geometry and algebra of vectors What is a vector ? (1) (Geometric vector) A vector is a directed line segment that corresponds to a displacement from one point A to another point B . We denote it by AB = v . (2) (Arithmetic or position vector) A vector is an ordered pair of real numbers; e.g. = = = 1 4 ) 1 , 4 ( 0 A a A vector such as A 0 with its initial point at the origin is said to be in standard position . In general, we define n (so called n -dimensional Euclidean vector space ) as the set of all ordered n -tuples of real numbers, written as () = = n n u u u u u u M L 2 1 2 1 , , u Vector algebra (1) Equality: Two vectors, ) , ( 2 1 u u = u and ) , ( 2 1 v v = v are equal if they have the same length and direction. u 1 =v 1 and u 2 =v 2 . one can be obtained by sliding (or translating) the other parallel to itself until the two vectors coincide. y b B A 0 x AB = v B A y x a Geometric vector Arithmetic vector
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___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 1-2 (2) Addition: Given two vectors, ) , ( 2 1 u u = u and ) , ( 2 1 v v = v , their addition, v u + is; ) , ( 2 2 1 1 v u v u + + = + v u ; ( component-wise addition ) parallelogram rule (3) Scalar multiplication: For a vector, ) , ( 2 1 u u = u , and a scalar c , the scalar multiple, c u is; ) , ( 2 1 cu cu c = u c u has the same direction as u if c > 0 and the opposite direction if c < 0 . Also, c u is c times as long as u . The combined arithmetic of the vector addition and the scalar multiplication is
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This note was uploaded on 06/06/2010 for the course MATH 54 taught by Professor Chung during the Spring '07 term at 카이스트, 한국과학기술원.

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LA07_1 - _ Linear Algebra: Spring 2007 Chapter 1. Vectors...

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