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Review of Finite Dimensional Vector Spaces

  1. Definition: A vector space is a set $ V$ with $ +$ and $ \cdot$ .

    \begin{gather*}\begin{array}{l} \mathbf{x}, \mathbf{y}\in V \\ a, b \in \Bbb{R} ...
...rrow \qquad a \cdot \mathbf{x}+ b \cdot \mathbf{y}= \mathbf{z}\in V\end{gather*}    

  2. Definition: Linear Combination

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

  3. Definition: $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ spans $ V$ , if

    $\displaystyle \mathbf{y}= \sum_{i=1}^N a_i \mathbf{x}_i \quad \forall \;\; \mathbf{y}\in V$    

  4. Definition: $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ are linearly independent if

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

    Otherwise they are linearly dependent .

  5. Definition: $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ is a basis for $ V$ if (1) they span $ V$ and (2) are linearly independent.

  6. Theorem: Let $ \{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ be a basis for $ V$ , then

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

    is unique.

  7. Definition: $ (a_1, a_2, \cdots, a_N)$ are the coordinates of $ \mathbf{y}$ with respect to the given basis for $ V$ .

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

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

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

    The coordinate vector ``lives'' in $ N$ -dimensional Euclidean space regardless of the nature of the vector space $ V$ .

Operations on Vectors in Euclidean Space
Let $ V$ be an $ N$ -dimensional Euclidean space and let $ \mathbf{a}, \mathbf{b}$ be elements of $ V$ .

  1. Definition: Inner Product

    $\displaystyle \langle \mathbf{a}, \mathbf{b}\rangle = \sum_{i=1}^N a_i b_i = \langle \mathbf{b}, \mathbf{a} \rangle$    Measures how alike or parallel $ \mathbf{a}$ and $ \mathbf{b}$ are.    

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

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

  3. Definition: $ \mathbf{a}$ and $ \mathbf{b}$ are orthogonal if $ \langle \mathbf{a}, \mathbf{b}\rangle = 0$ .

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

    $\displaystyle \Vert \mathbf{a}\Vert = \sqrt{ \langle \mathbf{a}, \mathbf{a}\rangle } = \sqrt{ \sum_{i=1}^N a_i^2 }$    Measures the length of $ \mathbf{a}$ .    
    $\displaystyle \Vert \mathbf{a}+ \mathbf{b}\Vert \leq \Vert \mathbf{a}\Vert + \V...
... \mathbf{b}\rangle \vert \leq \Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert$    

  5. Definition: Unit Vector

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

  6. Definition: Distance between two points $ \mathbf{a}, \mathbf{b}$

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

  7. Definition: $ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N$ are orthonormal if

    $\displaystyle \langle \mathbf{x}_i, \mathbf{x}_j \rangle = \left\{ \begin{array...
...xt{otherwise} \end{array}\right. = \delta_{i-j} \qquad \text{(Kronecker delta)}$    

Geometric Representation of Signals

Let $ V$ be an $ N$ -dimensional vector space of finite energy signals of duration $ T$ .

  1. Fact: Let $ \{x_1(t), x_2(t), \cdots, x_N(t)\}$ be a basis for $ V$ , then

    $\displaystyle y(t) = \sum_{i=1}^N a_i x_i(t) \qquad \qquad y(t) \leftrightarrow \mathbf{a} = \begin{bmatrix}a_1 & a_2 & \cdots & a_N \end{bmatrix}^T$    

  2. Definition: Inner product and norm of signals

    $\displaystyle \langle x_i(t), x_j(t) \rangle = \int_0^T x_i(t) x_j(t) dt$    
    $\displaystyle \Vert x_i(t) \Vert = \sqrt{ \langle x_i(t), x_i(t) \rangle } = \sqrt{ \int_0^T x_i(t) x_j(t) dt} = \sqrt{E_{x_i}}$    

  3. Fact: $ \{x_1(t), x_2(t), \cdots, x_N(t)\}$ are orthonormal means

    $\displaystyle \langle x_i(t), x_j(t) \rangle = \int_0^T x_i(t) x_j(t) dt = \lef...
...{ll} E_{x_i} = 1 & \text{if $i=j$,} \\ 0 & \text{otherwise} \end{array} \right.$    

  4. Definition: Unit energy signal

    $\displaystyle u(t) = \frac{y(t)}{\Vert y(t) \Vert} = \frac{y(t)}{\sqrt{E_y}} \qquad \qquad \Vert u(t) \Vert = \sqrt{E_u} = 1$    

  5. Representation: $ \{x_1(t), x_2(t), \cdots, x_N(t)\}$ is an orthonormal basis for $ V$ .

    $\displaystyle y(t)$ $\displaystyle \in V$ $\displaystyle y(t)$ $\displaystyle = \sum_{i=1}^N a_i x_i(t) \qquad a_i = \langle y(t), x_i(t) \rangle$    
    $\displaystyle y(t)$ $\displaystyle \not \in V$ $\displaystyle y(t)$ $\displaystyle = y_s(t) + y_n(t)$    
        $\displaystyle y_s(t)$ $\displaystyle = \sum_{i=1}^N a_i x_i(t) \qquad a_i = \langle y(t), x_i(t) \rangle$    
        $\displaystyle y_n(t)$ $\displaystyle = y(t) - y_s(t) \qquad \langle y_s(t), y_n(t) \rangle = 0$    

  6. Fact: Let $ y(t) \leftrightarrow \mathbf{a}$ and $ z(t)
\leftrightarrow \mathbf{b}$ .

$\displaystyle E_y = \int_0^T y^2(t) dt = \Vert y(t) \Vert^2 = \Vert \mathbf{a}\Vert^2 = \sum_{i=1}^N a_i^2$    
$\displaystyle \langle y(t) , z(t) \rangle = \int_0^T y(t) z(t) dt = \sum_{i=1}^N a_i b_i = \langle \mathbf{a}, \mathbf{b}\rangle$    
$\displaystyle d(y(t), z(t)) = \Vert y(t) - z(t) \Vert = \Vert \mathbf{a}- \mathbf{b}\Vert = d(\mathbf{a}, \mathbf{b})$    
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 Creative Commons License