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Jim Hefferon
linearalgebra
Commits
14812eef
Commit
14812eef
authored
Dec 05, 2016
by
Jim Hefferon
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graphics for slid three_ii
parent
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three_ii.tex
slides/three_ii.tex
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slides/three_ii.tex
View file @
14812eef
...
...
@@ 720,13 +720,17 @@ consider dropping the onetoone condition.
With that, there may be some output vectors
$
\vec
{
w
}
\in
W
$
for which there are many associated inputs,
$
\vec
{
v
}
\in
V
$
such that
$
h
(
\vec
{
v
}
)=
\vec
{
w
}$
.
This is a real difference.
Now, we can ask:
Now we can ask:
for any vector in the range
$
\vec
{
w
}
\in\rangespace
{
h
}$
what are the associated domain vectors
$
\vec
{
v
}
\in
V
$
?
\pause
\smallskip
\ExecuteMetaData
[../map2.tex]
{
InverseImage
}
\smallskip
The structure of the inverse image sets
will give us insight into the definition of homomorphism.
\end{frame}
...
...
@@ 787,8 +791,9 @@ This function $\map{h}{\Re^2}{\Re^2}$ is linear.
\begin{equation*}
\colvec
{
x
\\
y
}
\mapsto\colvec
{
x+y
\\
2x+2y
}
\end{equation*}
Here are elements of
$
h
^{

1
}
(
\colvec
{
1
\\
2
}
)
$
going to
$
\colvec
{
1
\\
2
}$
.
(Only one inverse image element is shown as a vector, most are shown as dots.)
Here are elements of
$
h
^{

1
}
(
\colvec
{
1
\\
2
}
)
$
.
(Only one inverse image element is shown as a vector, most are indicated
with dots.)
\begin{center}
\includegraphics
{
asy/three
_
ii
_
inv
_
img01.pdf
}
\quad\raisebox
{
0.25in
}{$
\longmapsto
$}
\quad
...
...
@@ 821,10 +826,14 @@ vector.
That is, preservation of addition is:
$
h
(
{
\color
{
red
}
\vec
{
v
}_
1
}
)+
h
(
{
\color
{
blue
}
\vec
{
v
}_
2
}
)
=
h
(
{
\color
{
magenta
}
\vec
{
v
}_
1
+
\vec
{
v
}_
2
}
)
$
.
\
pause
\end{frame}
\
begin{frame}
So the intuition is that a linear map organizes its domain into inverse
images, such that those sets reflect the structure of the range.
images,
\begin{center}
\includegraphics
{
../ch3.5
}
% bean to bean; many to one
\end{center}
such that those sets reflect the structure of the range.
\end{frame}
...
...
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