Chapter3 - Chapter 3 Vector Spaces (Euclidean n-Spaces)...

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Unformatted text preview: Chapter 3 Vector Spaces (Euclidean n-Spaces) Section 3.1 Euclidean n-Spaces Discussion 3.1.1 (Geometric Vectors) A vector can be represented geometrically as a directed line segment or an arrow; the direction of the arrow specifies the direction of the vector and the length of the arrow describes its magnitude. Two vectors are regarded as equal if they have the same length and direction. For example, in the following diagram, the two vectors u and v are the same but different from the vector w . a0 a0 a0 a18 u a0 a0 a0 a18 v a64 a64 a64 a82 w The addition, negative, difference and scalar multiple of vectors can be defined geomet- rically as follows. (a) The addition u + v of two vectors u and v : a45 u a0 a0 a0 a18 v a45 u a0 a0 a0 a18 v a16 a16 a16 a16 a16 a16 a16 a16 a16 a49 u + v Note that u + v is the same as v + u . (b) The negative − u of a vector u : a0 a0 a0 a18 u a0 a0 a0 a9 − u (c) The difference u − v of two vectors u and v : a45 u a0 a0 a0 a18 v a45 u a0 a0 a0 a9 − v a64 a64 a64 a82 u − v 2 Chapter 3. Vector Spaces (Euclidean n-Spaces) Note that u − v is the same as u + ( − v ). (d) The scalar multiple k u of a vector u ( k ∈ R ): a0 a0 a0 a18 u a0 a0 a18 1 2 u a0 a0 a0 a0 a0 a0 a18 2 u a0 a0 a0 a0 a0 a9 ( − 1 . 5) u Note that the negative − u of u is the same as the scalar multiple ( − 1) u . Discussion 3.1.2 (Coordinate Systems) 1. Vectors in 2-space: Suppose we position a vector u in the xy-plane such that its initial point is at the origin (0 , 0). The coordinates ( u 1 , u 2 ) of the end point of u are called the components of u and we write u = ( u 1 , u 2 ). (0 , 0) a45 x a54 y a0 a0 a0 a0 a0 a0 a0 a0 a18 u ( u 1 , u 2 ) u 1 u 2 (a) The addition of two vectors: Let u = ( u 1 , u 2 ) and v = ( v 1 , v 2 ). Then u + v = ( u 1 + v 1 , u 2 + v 2 ). (b) The scalar multiple of a vector: Let u = ( u 1 , u 2 ). Then c u = ( cu 1 , cu 2 ) for any real number c . 2. Vectors in 3-space: Similarly, a vector in the xyz-space can be represented by its three components on the x, y, z-axes: a45 y a54 z a0 a0 a0 a0 a0 a0 a9 x a1 a1 a1 a1 a1 a1 a1 a1 a1 a21 u ( u 1 , u 2 , u 3 ) u 3 u 2 a0 a0 a0 a0 a0 u 1 Section 3.1. Euclidean n-Spaces 3 Here we write u = ( u 1 , u 2 , u 3 ). (a) The addition of two vectors: Let u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ). Then u + v = ( u 1 + v 1 , u 2 + v 2 , u 3 + v 3 ). (b) The scalar multiple of a vector: Let u = ( u 1 , u 2 , u 3 ). Then c u = ( cu 1 , cu 2 , cu 3 ) for any real number c . Definition 3.1.3 An n-vector or ordered n-tuple of real numbers has the form ( u 1 , u 2 , . . . , u i , . . . , u n ) where u 1 , u 2 , . . . , u n are real numbers. The number u i in the i th position of an n-vector is called the i th component or the i th coordinate of the n-vector. Let u = ( u 1 , u 2 , . . . , u n ) and v = ( v 1 , v 2 , . . . , v n ) be two n-vectors....
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Chapter3 - Chapter 3 Vector Spaces (Euclidean n-Spaces)...

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