Page MenuHomec4science
Files F112644648

1_introlinalg.tex
No OneTemporary
Actions

File Metadata

Created
Sun, May 11, 22:27

1_introlinalg.tex
View Options

\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\usepackage{pst-3dplot}
\begin{document}
\begin{frame} \frametitle{Introduction to vector space and linear algebra}
You certainly know the mechanics already:
\begin{itemize}
\item ``standard'' (Euclidean) vectors, dot product
\item matrices, determinants, transposition, matrix-vector multiplication
\item change of basis, linear independence
\end{itemize}
\vspace{1em}
\pause
\vspace{1em}
What we're going to do today:
\begin{itemize}
\item highlight the practical (engineering) use of all that
\item appreciate how general and useful that framework is
\item try to think in multidimensional space
\end{itemize}
\vspace{1em}
\end{frame}
\begin{frame} \frametitle{Caveat}
\begin{center}
I can try to explain it to you but I cannot understand it for you ;)
\end{center}
\pause
\vspace{3em}
\begin{itemize}
\item abstract linear algebra is kinda dry
\item today's lecture is going to be densely packed and \textit{fast}
\item you will need to ``actively'' digest it at home
\end{itemize}
\end{frame}
\begin{frame} \frametitle{What is linear algebra}
\end{frame}
\begin{frame} \frametitle{What is algebra}
\centering
\textit{algebra is branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers}
\vspace{1em}
{\footnotesize [Britannica]}
\end{frame}
\begin{frame} \frametitle{What is linear algebra}
\textit{Linear} algebra:
\begin{itemize}
\item focuses on linear operations (addition and scaling)
\item operates on \textit{multidimensional} entities called vectors and matrices
\item defines a ``world'' called \textit{vector space} that has an intuitive geometrical structure
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Key ideas behind linear algebra}
\begin{itemize}
\item a vector can be ``anything''
\item any vector can be expressed as a linear combination of fundamental building blocks\\ (aka \textit{basis vectors})
\item basis vectors define a coordinate space
\item the ``amount'' of each basis vector in a given vector is the \textit{scalar} (numerical) value of the corresponding \textit{coordinates} in the space
\end{itemize}
\vspace{2em}
\pause
\begin{center}
and therefore:
\end{center}
\vspace{1ex}
\begin{itemize}
\item coordinates are just tuples of numbers ($\begin{bmatrix} 2 & 3.5 & 2 & -0.8 \end{bmatrix}^T$)
\item we can use standard linear algebra tools independently of the ``true'' nature of vectors
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Example}
\begin{center}
$\mathbb{R}^2: \qquad \mathbf{x} = \begin{bmatrix} x_0 \\ x_1 \end{bmatrix}$
\vspace{2em}
\psset{unit=10mm}
\begin{pspicture}(-4,-1)(8,4)
\psgrid[subgriddiv=1,gridlabels=0pt,gridcolor=gray,griddots=10](0,0)(-4,-1)(8,4)
\psaxes[labelFontSize=\scriptstyle]{->}(0,0)(-4,-1)(8,4)
\psline[linecolor=blue]{->}(1,0)\uput[-90](0.5,0){$\mathbf{e}_0$}
\psline[linecolor=green]{->}(0,1)\uput[-180](0,0.5){$\mathbf{e}_1$}
\only<2>{
\psline[linecolor=darkred]{->}(4,2)
\uput[30](4,2){$\mathbf{x} = \begin{bmatrix} 4 \\ 2 \end{bmatrix} = 4\mathbf{e}_0 + 2\mathbf{e}_1$}
}
\end{pspicture}
\end{center}
\end{frame}
\def\freqplot#1#2{
\begin{dspPlot}[width=2.5cm,height=0.7cm,xticks=none,yticks=1,sidegap=0]{0,1}{-1.3,1.3}
\moocStyle
\dspFunc[linewidth=1pt,linecolor=blue]{x 360 #1 2 mul 1 add mul mul sin}
\dspText(0.5,2){$\mathbf{v}_{#1} = \sin(#2 \pi t)$}
\end{dspPlot}\hspace{-2em}}
\begin{frame} \frametitle{Example}
\vspace{1em}
\centering
\begin{tabular}{cccc}
\freqplot{0}{}
&
\freqplot{1}{3}
&
\freqplot{2}{5}
\\
\freqplot{3}{7}
&
\freqplot{4}{9}
&
\freqplot{5}{11}
\end{tabular}
\[
\mathbf{x} = \myvector{1 & 1/3 & 1/5 & 1/7 & 1/9 & 1/11}^T
\]
\begin{dspPlot}[width=5cm,height=2cm,sidegap=0,xout=true]{-1,1}{-1.1,1.1}
\moocStyle
\dspFunc{% square wave expansion y = sum_n sin(2pi(2n+1)t)/(2n+1)
x 180 mul
0 % accumulator a
0 1 5 % start, inc, end for loop
{2 mul 1 add % 2n+1 = k
dup % stack: x a k k
3 index % x a k k x
mul sin exch div
add}
for
exch pop}
\end{dspPlot}
\end{frame}
\begin{frame} \frametitle{Example}
\centering
\includegraphics[height=6.5cm]{dogs.eps}
\end{frame}
\begin{frame} \frametitle{Analogy: LEGO}
\centering
basic building block: $\mathbf{g} = \myvector{1 & 1 & 1}^T$
\vspace{2em}
\includegraphics[height=4cm]{lego1.eps}
\end{frame}
\begin{frame} \frametitle{Analogy: LEGO}
\centering
reshaping: $\mathbf{x} = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix}\mathbf{g}$
\vspace{2em}
\includegraphics[height=4cm]{lego2.eps}
\end{frame}
\begin{frame} \frametitle{Analogy: LEGO}
\centering
combining: $\mathbf{y} = \mathbf{x} \oplus \mathbf{x}$
\vspace{2em}
\includegraphics[height=4cm]{lego3.eps}
\end{frame}
\begin{frame} \frametitle{Why use vector spaces in SP?}
\centering
all types of signals can be represented as vectors\\
in a suitable vector space
\end{frame}
\begin{frame} \frametitle{The unifying framework}
\begin{itemize}
\item space of continuous-time, finite-energy signals
\item space of continuous-time bandlimited signals
\item space of discrete-time signals of lenght $N$
\item space of discrete-time periodic signals of period $N$
\item etc.
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Main advantages}
Math will be easier and signal properties will automatically apply to all signal spaces:
\begin{itemize}
\item we can select the ``easiest'' space to prove a result
\item the Fourier Transform will be super easy to understand
\item and so will the proof of the sampling theorem
\end{itemize}
\vspace{1em}
Vector space is the closest thing to an algorithmic framework for SP:
\begin{itemize}
\item most vectors can be thought of as simple arrays
\item approximation and compression algorithms are simple applications of linear algebra
\item low-dimensional intuition is super helpful
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Are you into data science?}
\centering
\includegraphics[height=5.5cm]{separable.eps}
\end{frame}
\end{document}

Event Timeline

c4science · Help