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Vector Spaces

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Review :: Operations :: Geometric Representation

Operations on Vectors in Euclidean Space

In this section, let $ V$ be $ N$-dimensional Euclidean space and let $ \mathbf{a}, \mathbf{b}$ be elements of $ V$.

Definition: Inner Product

$\displaystyle \langle \mathbf{a}, \mathbf{b}\rangle = \sum_{i=1}^N a_i b_i = \langle \mathbf{b}, \mathbf{a} \rangle$    

In one sense, the inner product measures how alike or parallel two vectors are.

Definition: Norm of a vector $ \mathbf{a}$

$\displaystyle \Vert \mathbf{a}\Vert = \sqrt{ \langle \mathbf{a}, \mathbf{a}\rangle } = \sqrt{ \sum_{i=1}^N a_i^2 }$    

The norm measures the length of a vector. The norm satisfies the two inequalities

$\displaystyle \Vert \mathbf{a}+ \mathbf{b}\Vert \leq \Vert \mathbf{a}\Vert + \Vert \mathbf{b}\Vert$    
$\displaystyle \langle \mathbf{a}, \mathbf{b}\rangle \leq \vert \langle \mathbf{a}, \mathbf{b}\rangle \vert \leq \Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert$    

Definition: Angle between two vectors $ \mathbf{a}, \mathbf{b}$

$\displaystyle \langle \mathbf{a}, \mathbf{b}\rangle = \Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert \cos \theta$    

Two vectors are orthogonal if $ \langle \mathbf{a}, \mathbf{b}\rangle = 0$.

Definition: Unit Vector

$\displaystyle \mathbf{u}= \frac{\mathbf{a}}{\Vert \mathbf{a}\Vert} \qquad \qquad \Vert \mathbf{u}\Vert = 1$    

Definition: Distance between two vectors $ \mathbf{a}, \mathbf{b}$

$\displaystyle d(\mathbf{a}, \mathbf{b}) = \Vert \mathbf{a}-\mathbf{b}\Vert = \sqrt{ \sum_{i=1}^N (a_i - b_i)^2 }$    

Definition: The set of vectors $ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N$ are called orthonormal if

$\displaystyle \langle \mathbf{x}_i, \mathbf{x}_j \rangle = \left\{ \begin{array}{ll} 1 & \text{if $i=j$,} \\ 0 & \text{otherwise} \end{array} \right.$    
Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, June 29). Vector Spaces. Retrieved November 22, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Communication_Systems_I_1/vectors_spaces_2.htm. This work is licensed under a Creative Commons License. Creative Commons License
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