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# Vector Spaces

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Review ::  Operations  ::  Geometric Representation

## Review of Finite Dimensional Vector Spaces

Definition: An -dimensional vector space is a set of objects along with a definition of how to add two elements of 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 ( ).

Definition: Linear Combination

 or

Definition: A set of vectors in is said to span , if every vector in can be written as a linear combination of the elements in the set.

Definition: A set of vectors in is called linearly independent if the equation

holds only for . Otherwise they are linearly dependent .

Definition: A set of vectors in is called a basis for if they span and are linearly independent.

Theorem: If the set is a basis for then in the representation of

the coefficients are unique. The numbers are called the coordinates of with respect to the given basis for .

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

Definition: Let us construct the unique coordinate vector corresponding to the element in .

Note that the coordinate vector lives'' in -dimensional Euclidean space regardless of the nature of the vector space . For example, might be a vector space of polynomials. In digital communications, 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: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Communication_Systems_I_1/vectors_spaces_1.htm. This work is licensed under a Creative Commons License