Commit bb5636b0 authored by Jim Hefferon's avatar Jim Hefferon

three_iv

parent be4278d5
......@@ -55,27 +55,24 @@
\section{Sums and Scalar Products}
%..........
\begin{frame}{Representing operations on linear functions}
Recall that the collection of linear functions
between two spaces $\linmaps{V}{W}$ is a vector space.
That is, given
Recall that given
a linear map $\map{f}{V}{W}$ then
the scalar multiple $rf$
the scalar multiple function
\begin{equation*}
\vec{v}\mapsunder{r\cdot f} r\cdot (\,f(\vec{v})\,)
\end{equation*}
is a linear map $\map{rf}{V}{W}$.
And, where $\map{f,g}{V}{W}$ are
linear then their sum
And where $\map{f,g}{V}{W}$ are
linear then the function that is their sum
\begin{equation*}
\vec{v}\mapsunder{f+g} f(\vec{v})+g(\vec{v})
\end{equation*}
is also linear $\map{f+g}{V}{W}$.
\pause
We now see how the matrix representation of
$\rep{f}{B,D}$
is related to that of $\rep{rf}{B,D}$,
and how the representations of
and also how the representations of
$\rep{f}{B,D}$ and $\rep{g}{B,D}$
combine to give the representation of $\rep{f+g}{B,D}$.
\end{frame}
......@@ -83,15 +80,16 @@ combine to give the representation of $\rep{f+g}{B,D}$.
\ex
Fix a domain~$V$ and codomain~$W$ with bases
$B=\sequence{\vec{\beta}_1,\vec{\beta}_2}$
and $D=\sequence{\vec{\delta}_1,\vec{\delta}_2}$.
Let $\map{f}{V}{W}$ be the linear map represented
by a matrix~$H$
and $D=\sequence{\vec{\delta}_1,\vec{\delta}_2}$, and
let $\map{f}{V}{W}$ be the linear map represented
by a matrix~$H$.
We will find the matrix representing the map $\map{6f}{V}{W}$.
\begin{equation*}
\vec{v}\mapsunder{6f} 6\cdot f(\vec{v})
\end{equation*}
Let
this be the representation
\pause
Suppose that
this is the representation
of an output vector~$f(\vec{v})$.
\begin{equation*}
% \rep{\vec{v}}{B}=\colvec{v_1 \\ v_2}
......@@ -102,7 +100,7 @@ of an output vector~$f(\vec{v})$.
Note that
$6\cdot f(\vec{v})=6\cdot (w_1\vec{\delta}_1+w_2\vec{\delta}_2)
=(6w_1)\cdot\vec{\delta}_1+(6w_2)\cdot\vec{\delta}_2$,
so $6f\:(\vec{v})$ has
so the output vector $6f\:(\vec{v})$ has
this representation.
\begin{equation*}
% \rep{6f(\vec{v})}{D}
......@@ -140,7 +138,7 @@ and this is the matrix-vector representation of the action of~$6f$.
\colvec{v_1 \\ v_2}
=\colvec{12v_1+6v_2 \\ 18v_1+24v_2}
\end{equation*}
So this is the matrix representing~$6f$.
So this must be the matrix representing the function~$6f$.
\begin{equation*}
\rep{6f}{B,D}
=
......@@ -227,6 +225,10 @@ so this is its matrix representation.
\begin{equation*}
\rep{f+g}{B,D}=
\begin{mat}
2+5 &1+8 \\
3+7 &4+6
\end{mat}
=\begin{mat}
7 &9 \\
10 &10
\end{mat}
......@@ -274,7 +276,8 @@ Then
\end{mat}
\end{equation*}
None of these is defined: $A+B$, $B+A$,
$B+C$, $C+B$, because the sizes don't match.
$B+C$, $C+B$, because the sizes don't match,
making it impossible to do an entry-by-entry sum.
\end{frame}
......@@ -341,11 +344,11 @@ is composition.
\ExecuteMetaData[../map4.tex]{pf:CompositionOfLinearMapsIsLinear}
\qed
\pause
\medskip
We next do an exploratory calculation to see how the matrix
representations of the two functions combine to make
the matrix representation of their composition.
% \pause
% \medskip
% We next do an exploratory calculation to see how the matrix
% representations of the two functions combine to make
% the matrix representation of their composition.
\end{frame}
......@@ -374,13 +377,14 @@ represented as here.
% $V$ has dimension~$2$,
% $W$ has dimension~$3$,
% and $X$ has dimension~$2$.
We want to see how these two representations combine to
give the representation of the map
We will do an explatory computation,
to see how these two representations combine to
give the representation of the composition
$\map{\composed{g}{h}}{V}{X}$.
\pause
We start with the action
of~$h$ on a~$\vec{v}\in V$.
of~$h$ on~$\vec{v}\in V$.
\begin{align*}
\rep{h(\vec{v})}{C}
&=\rep{h}{B,C}\cdot\rep{\vec{v}}{B} \\
......@@ -394,11 +398,9 @@ of~$h$ on a~$\vec{v}\in V$.
=
\colvec{3v_1+v_2 \\ 2v_1+5v_2 \\ 4v_1+6v_2}
\end{align*}
Of course we represent application of~$h$ by doing the matrix vector
multiplication.
\end{frame}
\begin{frame}
\noindent Next apply $g$.
\noindent Next, to that apply $g$.
\begin{multline*}
\rep{g}{C,D}\cdot \rep{h(\vec{v})}{C}
=
......@@ -470,8 +472,7 @@ So here is how the two starting matrices combine.
\end{frame}
\begin{frame}
\ex
This product is not defined because
the number of columns on the left must equal the number of rows on the right.
This product
\begin{equation*}
\begin{mat}
1 &3 &-1 \\
......@@ -483,6 +484,8 @@ the number of columns on the left must equal the number of rows on the right.
2 &2 &0
\end{mat}
\end{equation*}
is not defined because
the number of columns on the left must equal the number of rows on the right.
\pause
\ex
......@@ -726,9 +729,9 @@ These two functions can be composed
\end{equation*}
because the codomain of~$f$ is the domain of~$g$.
Observe about the order of the notation:~in
An important observation about the order in which we write these things:~in
writing the composition~$\composed{g}{h}$,
the function $g$ is written first (meaning leftmost)
the function $g$ is written first, that is, leftmost,
but it is applied second.
\begin{equation*}
\vec{v}\mapsunder{h} h(\vec{v}) \mapsunder{g} g( h(\vec{v}))
......
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