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Vector Spaces Slides

Review of Finite Dimensional Vector Spaces

1. Definition: A vector space is a set with and .

2. Definition: Linear Combination

 or

3. Definition: spans , if

4. Definition: are linearly independent if

Otherwise they are linearly dependent .

5. Definition: is a basis for if (1) they span and (2) are linearly independent.

6. Theorem: Let be a basis for , then

is unique.

7. Definition: are the coordinates of with respect to the given basis for .

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

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

The coordinate vector lives'' in -dimensional Euclidean space regardless of the nature of the vector space .

Operations on Vectors in Euclidean Space
Let be an -dimensional Euclidean space and let be elements of .

1. Definition: Inner Product

 Measures how alike or parallel and are.

2. Definition: Angle between two vectors

3. Definition: and are orthogonal if .

4. Definition: Norm of a vector

 Measures the length of .

5. Definition: Unit Vector

6. Definition: Distance between two points

7. Definition: are orthonormal if

Geometric Representation of Signals

Let be an -dimensional vector space of finite energy signals of duration .

1. Fact: Let be a basis for , then

2. Definition: Inner product and norm of signals

3. Fact: are orthonormal means

4. Definition: Unit energy signal

5. Representation: is an orthonormal basis for .

6. Fact: Let and .

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 29). Vector Spaces Slides. 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_slides.htm. This work is licensed under a Creative Commons License