# c1s1 - 1 1.1 Vectors in Euclidean Space Chapter 1 Vectors...

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1.1 Vectors in Euclidean Space 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.1. Vectors in Euclidean Spaces Defnition. The space R n ,o r Euclidean n -space ,i se i th e r (1) the collection of all n -tuples of the form ( x 1 ,x 2 ,...,x n ) where the x i ’s are real numbers (the n -tuples are called points ), or (2) the collection of all n -tuples of the form [ x 1 2 n ] where the x i ’s are real numbers (the n -tuples are called vectors ). Note. There is as yet no diFerences between points and vectors. Note. R 1 is just the collection of real numbers (which we know to have an algebraic structure — addition and subtraction, say). R 2 is the collection of all points in the Cartesian plane. Notation. The book denotes vectors with bold faced letters. We use letters (usually lower case) with little arrows over them: ~x =[ x 1 2 n ] .

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1.1 Vectors in Euclidean Space 2 Defnition. For ~x R ,say ~x =[ x 1 ,x 2 ,...,x n ], the i th component of ~x is x i . Defnition. Two vectors in R n , ~v v 1 ,v 2 ,...,v n ]and ~w w 1 ,w 2 ,...,w n ] are equal if each of their components are equal. The zero vector , ~ 0, in R n is the vector of all zero components. Note. We have the following geometric interpretation of vectors: A vector R 2 can be drawn in standard position in the Cartesian plane by drawing an arrow from the point (0 , 0) to the point ( v 1 2 )whe re v 1 2 ]:
1.1 Vectors in Euclidean Space 3 We can draw ~v translated to point

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## This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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c1s1 - 1 1.1 Vectors in Euclidean Space Chapter 1 Vectors...

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