$page_title   = "Exhibit: Pascal Matrices";
$page_description = "Exhibit page in Math 21b, Linear Algebra and Applications";
${base}    = "../../";
$robots = "all";

sub main_page {
print OUTPUT<<"EOF"

<h1> Pascal Matrices</h1> 

In the <a href="../../final/final.pdf">Final exam</a>, we looked at
Pascal triangle matrices. So, here is the Pascal triangle, then the
Pascal triangle, in which the integers are seen modulo 2 (we look at
the remainder when dividing by 2, which means that we put 1 if the number
is odd and 0 if the number is even). It was generated with the following
line: 
<pre>
M=9; A=Table[ Binomial[n,k],{n,0,M-1},{k,0,M-1}]; Mod[A,2] 
</pre>

<center><table width="100%"><tr><td width="40%">
<pre>
1  0  0  0  0  0  0  0  0
1  1  0  0  0  0  0  0  0 
1  2  1  0  0  0  0  0  0 
1  3  3  1  0  0  0  0  0 
1  4  6  4  1  0  0  0  0 
1  5 10 10  5  1  0  0  0  
1  6 15 20 15  6  1  0  0 
1  7 21 35 35 21  7  1  0 
1  8 28 56 70 56 28  8  1 
</pre>
</td><td width="20%"> </td><td width="40%">
<pre>
1  0  0  0  0  0  0  0  0
1  1  0  0  0  0  0  0  0
1  0  1  0  0  0  0  0  0
1  1  1  1  0  0  0  0  0
1  0  0  0  1  0  0  0  0
1  1  0  0  1  1  0  0  0
1  0  1  0  1  0  1  0  0
1  1  1  1  1  1  1  1  0
1  0  0  0  0  0  0  0  1
</pre>
</td></tr></table></center>

Here is a larger version for n=2<sup>9</sup> = 512. The picture of the matrix
converges to a fractal called the 
<a href="https://en.wikipedia.org/wiki/Sierpinski_triangle">Sierpinski triangle</a>. <br>

<center>
<img src="pascal.png" width="100%">
<img src="pascalinverse.png" width="100%">
</center>

In the exam, we have computed the first column of the inverse of the Pascal matrix. 
Here is a picture of the inverse for n=2048. The inverse takes values 0,1,-1 only. 


EOF
}

1;