Personal tools
You are here: Home → Electrical and Computer Engineering → Communication Systems I → Vector Spaces

Vector Spaces

Document Actions
  • Send this
  • Print this
  • Content View
  • Bookmarks
  • CourseFeed
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 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_1.htm. This work is licensed under a Creative Commons License. Creative Commons License
Reuse Course
Download this course