Vector Spaces Slides

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Review of Finite Dimensional Vector Spaces

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

$\begin{gather*}\begin{array}{l} \mathbf{x}, \mathbf{y}\in V \\ a, b \in \Bbb{R} ... ...rrow \qquad a \cdot \mathbf{x}+ b \cdot \mathbf{y}= \mathbf{z}\in V\end{gather*}$
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: $\{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ spans , if

$\displaystyle \mathbf{y}= \sum_{i=1}^N a_i \mathbf{x}_i \quad \forall \;\; \mathbf{y}\in V$
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.
Definition: $\{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ is a basis for if (1) they span and (2) are linearly independent.
Theorem: Let $\{ \mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_N\}$ be a basis for , then

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

is unique.
Definition: $(a_1, a_2, \cdots, a_N)$ are the coordinates of $\mathbf{y}$ with respect to the given basis for .
Definition: The dimension of is the number of elements in a basis for .
Definition: Let us construct the unique $N \times 1$ coordinate vector corresponding to the element $\mathbf{y}$ in .

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

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 $\mathbf{a}, \mathbf{b}$ be elements of

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

$\displaystyle \langle \mathbf{a}, \mathbf{b}\rangle = \Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert \cos \theta$
Definition: $\mathbf{a}$ and $\mathbf{b}$ are orthogonal if $\langle \mathbf{a}, \mathbf{b}\rangle = 0$ .
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 $\mathbf{a}$ .

$\displaystyle \Vert \mathbf{a}+ \mathbf{b}\Vert \leq \Vert \mathbf{a}\Vert + \V... ... \mathbf{b}\rangle \vert \leq \Vert \mathbf{a}\Vert \cdot \Vert \mathbf{b}\Vert$
Definition: Unit Vector

$\displaystyle \mathbf{u}= \frac{\mathbf{a}}{\Vert \mathbf{a}\Vert} \qquad \qquad \Vert \mathbf{u}\Vert = 1$
Definition: Distance between two points $\mathbf{a}, \mathbf{b}$

$\displaystyle d(\mathbf{a}, \mathbf{b}) = \Vert \mathbf{a}-\mathbf{b}\Vert = \sqrt{ \sum_{i=1}^N (a_i - b_i)^2 }$
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 be an -dimensional vector space of finite energy signals of duration .

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

$\displaystyle y(t) = \sum_{i=1}^N a_i x_i(t) \qquad \qquad y(t) \leftrightarrow \mathbf{a} = \begin{bmatrix}a_1 & a_2 & \cdots & a_N \end{bmatrix}^T$
Definition: Inner product and norm of signals

$\displaystyle \langle x_i(t), x_j(t) \rangle = \int_0^T x_i(t) x_j(t) dt$

$\displaystyle \Vert x_i(t) \Vert = \sqrt{ \langle x_i(t), x_i(t) \rangle } = \sqrt{ \int_0^T x_i(t) x_j(t) dt} = \sqrt{E_{x_i}}$
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.$
Definition: Unit energy signal

$\displaystyle u(t) = \frac{y(t)}{\Vert y(t) \Vert} = \frac{y(t)}{\sqrt{E_y}} \qquad \qquad \Vert u(t) \Vert = \sqrt{E_u} = 1$
Representation: $\{x_1(t), x_2(t), \cdots, x_N(t)\}$ is an orthonormal basis for .

$\displaystyle y(t)$ $\displaystyle \in V$ $\displaystyle y(t)$ $\displaystyle = \sum_{i=1}^N a_i x_i(t) \qquad a_i = \langle y(t), x_i(t) \rangle$

$\displaystyle y(t)$ $\displaystyle \not \in V$ $\displaystyle y(t)$ $\displaystyle = y_s(t) + y_n(t)$

$\displaystyle y_s(t)$ $\displaystyle = \sum_{i=1}^N a_i x_i(t) \qquad a_i = \langle y(t), x_i(t) \rangle$

$\displaystyle y_n(t)$ $\displaystyle = y(t) - y_s(t) \qquad \langle y_s(t), y_n(t) \rangle = 0$
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 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_slides.htm. This work is licensed under a Creative Commons License.

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Vector Spaces Slides

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Review of Finite Dimensional Vector Spaces

Geometric Representation of Signals