two_i_a.tex

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\title[Spaces and their subspaces] % (optional, use only with long paper titles)
{Spaces and their subspaces}

\author{\textit{Linear Algebra} \\ {\small Jim Hef{}feron}}
\institute{
  \texttt{http://joshua.smcvt.edu/linearalgebra}
}
\date{}


\subject{Spaces and their subspaces}
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\begin{document}
\begin{frame}
  \titlepage
\end{frame}

% =============================================

% ..... Two.I.2.19 .....
\section{Real three-space}
\begin{frame}{Subspaces of $\Re^3$}
The next slide gives a sample of subspaces of the vector space~$\Re^3$:
the entire space, planes, lines, and the trivial subspace.
Subsets are drawn connected to their supersets on the levels 
directly above and below.

(On the level one up from the bottom, the second 
and third subspaces are lines because
the conjunction of the two conditions means that each is the 
intersection of two planes.)
\end{frame}

\begin{frame}
{\centering\includegraphics{asy/r3_subspaces.pdf}}  
\end{frame}


\begin{frame}{Express subspaces as spans}
\ex Consider the plane.
\begin{equation*}
  P=\set{\colvec{x \\ y \\ z}\suchthat x+y+z=0}
\end{equation*}
Take the condition $x+y+z=0$ as a one-equation linear system
and parametrize.
\begin{equation*}
  P=\set{\colvec{-y-z \\ y \\ z}\suchthat y,z\in\Re}
   =\set{\colvec{-1 \\ 1 \\ 0}y+\colvec{-1 \\ 0 \\ 1}z\suchthat y,z\in\Re}
\end{equation*}
Thus the plane is this span.
\begin{equation*}
  P=\spanof{\,\set{\colvec{-1 \\ 1 \\ 0}, 
            \colvec{-1 \\ 0 \\ 1}}\,}
\end{equation*}
\end{frame}
\begin{frame}
This shows~$P$.
\begin{center}
  \includegraphics{asy/two_i_a_plane.pdf}
\end{center}
The two vectors from the spanning set are in red.
For each, its body lies in the plane.
\end{frame}



\begin{frame}
\ex 
For the plane
\begin{equation*}
  \hat{P}=\set{\colvec{x \\ y \\ z}\suchthat x+2z=0}
\end{equation*}
repeat the process
\begin{equation*}
  \hat{P}=\set{\colvec{-2z \\ y \\ z}\suchthat y,z\in\Re}
         =\set{\colvec{0 \\ 1 \\ 0}y+\colvec{-2 \\ 0 \\ 1}z\suchthat y,z\in\Re}
\end{equation*}
to express it as a span.
\begin{equation*}
  \hat{P}=\spanof{\,\set{\colvec{0 \\ 1 \\ 0},
                         \colvec{-2 \\ 0 \\ 1}}\,}
\end{equation*}

\pause
\ex
The $xy$-plane is a span in a natural way.
\begin{equation*}
  \text{$xy$~plane}=\spanof{\,\set{\colvec{1 \\ 0 \\ 0},
                                   \colvec{0 \\ 1 \\ 0}}\,}
\end{equation*}
\end{frame}



\begin{frame}
\ex
Next, parametrize the lines.
The conditions in the set description
\begin{equation*}
  L=\set{\colvec{x \\ y \\ z}\suchthat \text{$x-y+z=0$ and $x+2z=0$}}
\end{equation*}
make a linear system.
\begin{equation*}
  \begin{linsys}{3}
    x &- &y &+ &z  &= &0 \\
    x &  &  &+ &2z &= &0
  \end{linsys}
  \grstep{-\rho_1+\rho_2}
  \begin{linsys}{3}
    x &- &y &+ &z  &= &0 \\
      &  &y &+ &z  &= &0
  \end{linsys}
\end{equation*}
Parametrizing
\begin{equation*}
  L=\set{\colvec{x \\ y \\ z}\suchthat \text{$y=-z$ and $x=-2z$}}
\end{equation*}
gives this.
\begin{equation*}
  L=\set{\colvec{-2 \\ -1 \\ 1}z\suchthat z\in\Re}
   =\spanof{\,\set{\colvec{-2 \\ -1 \\ 1}}\,}
\end{equation*}
\end{frame}
\begin{frame}
Here is the line, drawn with the same view as the earlier plane.
\begin{center}
  \includegraphics{asy/two_i_a_line.pdf}
\end{center}
In red is the spanning vector
\begin{equation*}
  \colvec{-2 \\ -1 \\ 1}
\end{equation*}
(its endpoint lies behind the plane of the screen).
\end{frame}


\begin{frame}
\ex
The line
\begin{equation*}
  \hat{L}=\set{\colvec{x \\ y \\ z}\suchthat \text{$y=2x$ and $z=0$}}
\end{equation*}
is easy to describe in a parametrized way.
\begin{align*}
  \hat{L} 
  &=\set{\colvec{1/2 \\ 1 \\ 0}y\suchthat y\in\Re}  \\
  &=\spanof{\,\set{\colvec{1/2 \\ 1 \\ 0}}\,}
\end{align*}

\pause
\ex
The $y$-axis is also easy to describe as a span.
\begin{equation*}
  \text{$y$-axis}
  =\spanof{\,\set{\colvec{0 \\ 1 \\ 0}}\,}
\end{equation*}
\end{frame}


\begin{frame}
\ex 
We can describe the entire space as a span.
\begin{equation*}
  \Re^3=\spanof{\,\set{\colvec{1 \\ 0 \\ 0},
                       \colvec{0 \\ 1 \\ 0},
                       \colvec{0 \\ 0 \\ 1}}\,}
\end{equation*}

\pause
\ex
We can do the same for the trivial subspace.
\begin{equation*}
  \set{\colvec{0 \\ 0 \\ 0}}
  =\spanof{\,\set{}\,}
\end{equation*}
(Remember the convention that a sum of zero-many members of $\Re^n$ 
is the zero vector.)
\end{frame}

\begin{frame}{$\Re^3$'s diagram reprised}
The following slide repeats the diagram of $\Re^3$'s subspaces, 
showing the same subspaces.
On this diagram those subspaces they are described as spans, 
where the spanning set uses a minimal number of vectors.

By `minimal' we mean something like:~while 
we could describe the $xy$-plane in either of these ways,
\begin{equation*}
  \text{$xy$-plane}
  =\spanof{\,\set{\colvec{1 \\ 0 \\ 0},
                  \colvec{0 \\ 1 \\ 0}}\,}
  =\spanof{\,\set{\colvec{1 \\ 0 \\ 0},
                  \colvec{0 \\ 1 \\ 0},
                  \colvec{2 \\ -1 \\ 0}}\,}
\end{equation*}
the second has an extra vector, so the next slide doesn't use that description.
(We will soon make this precise.)

% \pause\medskip
% Note that 
% the subspaces fall naturally into levels, 
% depending on how many vectors are in a minimal spanning set.
\end{frame}
\begin{frame}
{\centering\includegraphics{asy/r3_subspaces_spans.pdf}}
\end{frame}




%...........................
\section{The space of quadratic polynomials}
\begin{frame}{$\polyspace_2$'s subspaces}
The next slide has a picture of some of the subspaces of the
space of quadratic polynomials.
As with the $\Re^3$ diagram, 
subsets are shown connected to supersets on the level above.

With $\Re^3$ the geometry gave us a good start for a natural classification
of subspaces into planes, lines, etc.
In this space there is less of that sense.
But there are a couple of natural subspaces.
\begin{align*}
  \text{linears}
    &=\set{0x^2+bx+c\suchthat b,c\in\Re}  \\
  \text{constants}
    &=\set{0x^2+0x+c\suchthat c\in\Re}  
\end{align*}
These are on the next slide
(which is why they have abbreviated names here, to fit in the layout), 
along with a couple of more generic spaces.
\end{frame}
\begin{frame}
{\centering\includegraphics{asy/p2_subspaces.pdf}}  
\end{frame}


\begin{frame}{Parametrizing}
\ex
For the top line, this will do.
\begin{equation*}
  \polyspace_2
  =\spanof{\,\set{x^2,x,1}\,}
  =\spanof{\,\set{x^2+0x+0,
               0x^2+x+0,
               0x^2+0x+1}\,}       
\end{equation*}

\ex
The trivial subspace is the span of the empty set.
\begin{equation*}
  \set{0}
  =\set{0x^2+0x+0}
  =\spanof{\,\set{}\,}
\end{equation*}

\pause
\ex
The set of linear polynomials has a natural expression as a span.
\begin{equation*}
  \text{linears}
  =\set{bx+c\suchthat b,c\in\Re}
  =\spanof{\,\set{x,1}\,}
\end{equation*}

\ex
The set of constant polynomials is similar.
\begin{equation*}
  \text{constants}
  =\set{0x^2+0x+c\suchthat c\in\Re}
  =\spanof{\,\set{1}\,}
\end{equation*}
\end{frame}


\begin{frame}
\ex
Paramatrize $P=\set{ax^2+bx+c\suchthat a+b+c=0}$ 
in much the same way as for subspaces of real three-space.
\begin{align*}
  % \set{ax^2+bx+c\suchthat a+b+c=0}
  P
  &=\set{ax^2+bx+c\suchthat a=-b-c}     \\
  &=\set{(-b-c)\cdot x^2+bx+c\suchthat b,c\in\Re}  \\     
  &=\set{(-x^2+x)\cdot b+(-x^2+1)\cdot c\suchthat b,c\in\Re}       
\end{align*}
The spanning set is $\set{-x^2+x, -x^2+1}$.

\pause
\ex
Simliarly for 
$\hat{P}=\set{ax^2+bx+c\suchthat \text{$a-c=0$ and $b-c=0$}}$.
\begin{align*}
  \hat{P}
  &=\set{ax^2+bx+c\suchthat \text{$a=c$ and $b=c$}}     \\
  &=\set{cx^2+cx+c\suchthat c\in\Re}     \\
  &=\set{(x^2+x+1)\cdot c \suchthat c\in\Re}       
\end{align*}
The spanning set is $\set{x^2+x+1}$.

\pause\medskip
The next slide reprises $\polyspace_2$'s diagram,
with the subspaces described as spans.
\end{frame}
\begin{frame}
{\centering\includegraphics{asy/p2_subspaces_spans.pdf}}  
\end{frame}


\section{Putting the examples together}
\begin{frame}
  \frametitle{Summary}

Subspaces are naturally described as spans.
In both examples these spans fall naturally into levels, 
according to the 
number of elements in a minimal spanning set. 

The book's next section gives a precise definition of 
when a spanning set is `minimal'.
The section after that shows that for any space, 
two minimal spanning sets have the same number of vectors.
\end{frame}

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