Skip to content

Commit

Permalink
feat: add proofs relating to set operations and images
Browse files Browse the repository at this point in the history
  • Loading branch information
@cuppajoeman
cuppajoeman committed Sep 8, 2024
1 parent 2dc03b6 commit ec20f2a
Show file tree
Hide file tree
Showing 13 changed files with 1,073 additions and 577 deletions.
0 ...complex/algebra_of_the_complex_plane.html → ...complex/algebra_of_the_complex_plane.html
View file
File renamed without changes.
0 html/complex/complex.htm → html/algebra/complex/complex.htm
View file
File renamed without changes.
221 changes: 221 additions & 0 deletions html/algebra/complex/complex_numbers.html
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,221 @@
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<title>Complex Numbers</title>
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="/styles/styles.css">

<script src="/js/script.js" defer></script>

</head>

<body>
<div class="thin-wrapper">

<div class="definition" id="definition-addition">
<div class="title">Addition</div>
<div class="content">
Given \( a + b i , c + d i \in \mathbb{C} \), we define
\[
\left ( a + b i \right ) + \left ( c + d i \right ) = \left ( a + c \right ) + \left ( b + d \right ) i \
\]
</div>
</div>

<div class="definition" id="definition-multiplication">
<div class="title">Multiplication</div>
<div class="content">
given \( a + b i , c + d i \in \mathbb{C} \), then:
\[ \left ( a + b i \right ) \cdot \left ( c + d i \right ) = a b + i \left ( a d + b c \right ) - d b = \left ( a d - b d \right ) + i \left ( a d + b c \right ) \]
</div>
</div>

<div class="definition" id="definition-equality">
<div class="title">Equality</div>
<div class="content">
Given \( a + b i , c + d i \in \mathbb{C} \), they are equal when \( a = c \) and \( b = d \)
</div>
</div>

<div class="definition" id="definition-complex-conjugate">
<div class="title">Conjugate</div>
<div class="content">
suppose \( z = x + i y \), then the conjugate is defined and denoted by
\[
\overline{z} := x - i y
\]
</div>
</div>

<div class="proposition" id="proposition-conjugate-cancels">
<div class="title">conjugate Cancels</div>
<div class="content">
\( \overline{\overline{z}} = z \)
</div>
<div class="proof">
Let \( z = x + i y \), then \( \overline{z} = x - i y = x + i \left ( - y \right ) \) then \( \overline{\overline{z}} = x - i \left ( - y \right ) = x + i y = z \) as needed.
</div>
</div>

<div class="definition" id="definition-modulus">
<div class="title">Modulus</div>
<div class="content">
for any \( z = x + i y \in \mathbb{C} \), we define
\[ \left | z \right | := \sqrt{x^{2} + y^{2}} \]
</div>
</div>

<div class="corollary" id="corollary-modulus-ignores-conjugates">
<div class="title">Modulus Ignores Conjugates</div>
<div class="content">
\[ \left | \overline{z} \right | = \left | z \right | \]
</div>
<div class="proof">
Suppose that \( z = x + i y \), then \( \left | \overline{z} \right | = \left | x - i y \right | = \sqrt{x^{2} + \left ( - y \right )^{2}} = \sqrt{x^{2} + y^{2}} = \left | z \right | \)
</div>
</div>




<div class="proposition" id="proposition-complex-number-times-its-conjugate-equals-the-modulus-squared">
<div class="title">complex number times it's conjugate equals the modulus squared</div>
<div class="content">
<div class="centered-content">
\( z \cdot \overline{z} = \left \lvert z \right \rvert^{2} \)
</div>
</div>
<div class="proof">
suppose that \( z = x + i y \), then \( z \cdot \overline{z} = \left ( x + i y \right ) \cdot \left ( x - i y \right ) = x^{2} + y^{2} + x y i - x y i = x^{2} + y^{2} \), but then again \( \left | z \right | = \left | x + i y \right | = \sqrt{x^{2} + y^{2}} \) thus \( z \cdot \overline{z} = \left | z \right | ^{2} \)
</div>
</div>



<div class="definition" id="definition-inverse">
<div class="title">inverse</div>
<div class="content">
the inverse of a complex number \( z \in \mathbb{C} \) is another \( w \in \mathbb{C} \) such that \( z \cdot w = 1 \), we denote \( w \) by \( z^{- 1} \) or \( \frac{1}{z} \)
</div>
</div>

<div class="theorem" id="theorem-value-of-the-inverse">
<div class="title">value of the inverse</div>
<div class="content">
Given \( z \in \mathbb{C} \) we have \( z^{- 1} = \frac{\overline{z}}{\left \lvert z \right \rvert^{2}} \)
</div>
<div class="proof">
We <span class="knowledge-link" data-href="/complex/algebra_of_the_complex_plane.html#proposition-complex-number-times-its-conjugate-equals-the-modulus-squared">know</span> that \( z \cdot \overline{z} = \left \lvert z \right \rvert^{2} \), therefore \( z \cdot \frac{\overline{z}}{\left \lvert z \right \rvert^{2}} = 1 \), so by the <span class="knowledge-link" data-href="/complex/algebra_of_the_complex_plane.html#definition-inverse">definition of inverse</span> \( z^{- 1} = \frac{\overline{z}}{\left \lvert z \right \rvert^{2}} \) as needed.
</div>
</div>



<div class="exercise">
<div class="title"></div>
<div class="content">
For any \( z \in \mathbb{C} , \left \lvert \frac{z}{\overline{z}} \right \rvert = 1 \)
</div>
<div class="proof">
<p>
By the <span class="knowledge-link" data-href="/complex/algebra_of_the_complex_plane.html#theorem-value-of-the-inverse">value of the inverse</span> we have: \( \frac{1}{\overline{z}} = \frac{\overline{\overline{z}}}{\left \lvert \overline{z} \right \rvert^{2}} \) since we know that <span class="knowledge-link" data-href="/complex/algebra_of_the_complex_plane.html#proposition-conjugate-cancels">two complex conjugates cancel</span> and <span class="knowledge-link" data-href="/complex/algebra_of_the_complex_plane.html#corollary-modulus-ignores-conjugates">modulus ignores conjugates</span>, then \( \frac{\overline{\overline{z}}}{\left \lvert \overline{z} \right \rvert^{2}} = \frac{z}{\left \lvert z \right \rvert^{2}} \)
</p>
<p>
Now \( \left | \frac{z}{\overline{z}} \right | = \left | z \left ( \frac{z}{\left | z \right | ^{2}} \right ) \right | = \left | \left ( \frac{z}{\left | z \right | } \right )^{2} \right | = \left | \frac{z}{\left | z \right | } \right | \cdot \left | \frac{z}{\left | z \right | } \right | = 1 \cdot 1 = 1 \) as needed
</p>
</div>
</div>

<div class="definition" id="definitition-real-part-of-a-complex-number">
<div class="title">real part of a complex number</div>
<div class="content">
given \( z = x + i y \in \mathbb{C} \) we say that \( x \) is the real part of \( z \) and define the function
<div class="centered-content">
\( \mathcal{R} \left ( z \right ) = x \)
</div>
</div>
</div>

<div class="definition" id="definition-imaginary-part-of-a-complex-number">
<div class="title">Imaginary part of a Complex Number</div>
<div class="content">
given \( z = x + i y \in \mathbb{C} \) we say that \( y \) is the imaginary part of \( z \) and define the function
<div class="centered-content">
\( \mathcal{I} \left ( z \right ) = y \)
</div>
</div>
</div>

<div class="proposition" id="proposition-extracting-the-real-part-of-a-complex-number">
<div class="title">extracting the real part of a complex number</div>
<div class="content">
suppose \( z \in \mathbb{C} \), then \( \mathfrak{R} \left ( z \right ) = \frac{z + \overline{z}}{2} \)
</div>
<div class="proof"></div>
</div>


<div class="proposition" id="proposition-modulus-is-greater-than-it's-components">
<div class="title">modulus is greater than it's components</div>
<div class="content">
for any \( z \in \mathbb{C} \) we have both
<ul>
<li>\( \left | \mathfrak{R} \left ( z \right ) \right | \le \left | z \right | \)</li>
<li>\( \left | \mathfrak{I} \left ( z \right ) \right | \le \left | z \right | \)</li>
</ul>
</div>
<div class="proof"></div>
</div>


<div class="proposition" id="proposition-imaginary-part-distributes">
<div class="title">imaginary part distributes</div>
<div class="content">
\( \mathfrak{I} \left ( z + w \right ) = \mathfrak{I} \left ( z \right ) + \mathfrak{I} \left ( w \right ) \)
</div>
<div class="proof"></div>
</div>

<div class="proposition" id="proposition-modulus-plus-one-of-it's-components-is-positive">
<div class="title">modulus plus one of it's components is positive</div>
<div class="content">
Suppose that \( f{\in} \left \lbrace \mathcal{R} , \mathcal{I} \right \rbrace \), then
<div class="centered-content">
\( \left | z \right | + f{\left ( z \right )} \ge 0 \)
</div>
</div>
<div class="proof">
Let \( z = x + i y \in \mathbb{C} \) then suppose without loss of generality that \( f{=} \mathcal{R} \), then
<div class="centered-content">
\( - x \le \left | x \right | = \sqrt{x^{2}} \le \sqrt{x^{2} + y^{2}} = \left | x + i y \right | \)
</div>
therefore \( 0 \le x + \left | x + i y \right | \) which means \( 0 \le \mathcal{R} \left ( z \right ) + \left | z \right | \)
</div>
</div>


<div class="exercise" id="exercise-creating-squares">
<div class="title">creating squares</div>
<div class="content">
without use of the triangle inequality prove that \( \left | z - 1 \right | ^{2} = \left | z \right | ^{2} + 1 - 2 \mathcal{R} \left ( z \right ) \), then conclude that \( \left | z - 1 \right | \le \left | z \right | + 1 \)
</div>
<div class="proof">
<p>
\( \left | z - 1 \right | ^{2} \) <a href="#a-complex-number-times-it's-conjugate-equals-the-modulus-squared">=</a> \( \left ( z - 1 \right ) \cdot \overline{z - 1} = \left ( z - 1 \right ) \cdot \left ( \overline{z} - 1 \right ) = z \cdot \overline{z} - z - \overline{z} + 1 \), now recall that \( \mathcal{R} \left ( z \right ) = \frac{z + \overline{z}}{2} \) so we can continue the equality with \( \left | z \right | ^{2} - 2 \mathcal{R} \left ( z \right ) + 1 \), showing the first part
</p>
<p>
for the second part, we can notice that \( \left ( \left | z \right | + 1 \right )^{2} = \left | z \right | ^{2} + 2 \left | z \right | + 1 \), so then \( \left | z - 1 \right | ^{2} = \left | z \right | ^{2} + 1 - 2 \mathcal{R} \left ( z \right ) = \left | z \right | ^{2} + 2 \left | z \right | - 2 \left | z \right | + 1 - 2 \mathcal{R} \left ( z \right ) = \left ( \left | z \right | + 1 \right )^{2} - 2 \left ( \left | z \right | + \mathcal{R} \left ( z \right ) \right ) \), then by
<a href="#modulus-plus-one-of-it's-components-is-positive">this</a> proposition we can see that the last term in the previous equality chain is less than or equal to \( \left ( \left | z \right | + 1 \right )^{2} \) allowing us to conclude that \( \left | z - 1 \right | ^{2} \le \left ( \left | z \right | + 1 \right )^{2} \) which implies that \( \left | z - 1 \right | \le \left | z \right | + 1 \)
</p>
</div>



</div>

</body>

</html>
0 html/complex/images/higher_order_poles.png → ...bra/complex/images/higher_order_poles.png
View file
File renamed without changes
3 changes: 3 additions & 0 deletions html/algebra/linear/affine.txt
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,3 @@
homogenius coordinates, when you represent 2d information in 3d space so that we can do translations of 2d objects using linear algebra


2 changes: 1 addition & 1 deletion html/algebra/linear/index.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -22,7 +22,7 @@
<li><a href="matrices.html">Matrices</a></li>
<li><a href="systems_of_equations.html">Systems of Equations</a></li>
<li><a href="vector_spaces">Vector Spaces</a></li>
<li><a href="vector_spaces/linear_transformations.html">Vector Spaces</a></li>
<li><a href="vector_spaces/linear_transformations.html">Linear Transformations</a></li>
<li>polynomials</li>
<li>eigenvectors</li>
<li>determinants</li>
Expand Down
15 changes: 14 additions & 1 deletion html/algebra/linear/matrices.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -350,7 +350,20 @@ <h1>Matrices</h1>
</div>
</div>



<div class="definition" id="definition-hermitian-matrix">
<div class="title">Hermitian Matrix</div>
<div class="content">
A matrix \( A \in M _ { n \times n } \left( \mathbb{ C } \right) \) is said to be <b>hermitian</b> diff
\[
A = {\bar A} ^ T
\]
where we've applied
</div>
</div>

<p>
</p>



Expand Down
30 changes: 30 additions & 0 deletions html/algebra/linear/the_geometry_of_linear_transformations.html
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,30 @@
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">

<title>The Geometry of Linear Transformations</title>

<link rel="stylesheet" href="/styles/styles.css">
<script src="/js/script.js" defer></script>

</head>
<body>

<div class="thin-wrapper">

<div class="definition" id="definition-">
<div class="title"></div>
<div class="content">
</div>
</div>

</div>



<script src="" async defer></script>
</body>
</html>
Loading

0 comments on commit ec20f2a

Please sign in to comment.