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Vector Spaces

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Review ::  Operations  ::  Geometric Representation

Review of Finite Dimensional Vector Spaces

Definition: An $ N$ -dimensional vector space is a set $ V$ of objects along with a definition of how to add two elements of $ V$ and a definition of how to scale an element by a real number. The example to keep in mind is the one you are already familiar with-three dimensional Euclidean space ( $ N=3$ ).

Definition: Linear Combination

$\displaystyle \mathbf{y}= a_1 \mathbf{x}_1 + a_2 \mathbf{x}_2 + \cdots + a_N \mathbf{x}_N$    or $\displaystyle \qquad \mathbf{y}= \sum_{i=1}^N a_i \mathbf{x}_i$    

Definition: A set $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ of vectors in $ V$ is said to span $ V$ , if every vector $ \mathbf{y}$ in $ V$ can be written as a linear combination of the elements in the set.

Definition: A set $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ of vectors in $ V$ is called linearly independent if the equation

$\displaystyle 0 = a_1 \mathbf{x}_1 + a_2 \mathbf{x}_2 + \cdots + a_N \mathbf{x}_N$    

holds only for $ a_1=a_2=\cdots=a_N=0$ . Otherwise they are linearly dependent .

Definition: A set $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ of vectors in $ V$ is called a basis for $ V$ if they span $ V$ and are linearly independent.

Theorem: If the set $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ is a basis for $ V$ then in the representation of $ \mathbf{y}$

$\displaystyle \mathbf{y}= a_1 \mathbf{x}_1 + a_2 \mathbf{x}_2 + \cdots + a_N \mathbf{x}_N$    

the coefficients $ a_1, a_2, \cdots, a_N$ are unique. The numbers $ a_i$ are called the coordinates of $ \mathbf{y}$ with respect to the given basis for $ V$ .

Definition: The dimension of a vector space $ V$ is the number of elements in a basis for $ V$ .

Definition: Let us construct the unique $ N \times 1$ coordinate vector corresponding to the element $ \mathbf{y}$ in $ V$ .

$\displaystyle \begin{bmatrix}a_1 \\ a_2 \\ \vdots \\ a_N \end{bmatrix}$    

Note that the coordinate vector ``lives'' in $ N$ -dimensional Euclidean space regardless of the nature of the vector space $ V$ . For example, $ V$ might be a vector space of polynomials. In digital communications, $ V$ is a space of functions.
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 29). Vector Spaces. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: This work is licensed under a Creative Commons License Creative Commons License