Applied edits

Advanced/Linear Algebra 101/main.tex

@@ -43,8 +43,8 @@
 	\problem{}
 	Show that the euclidean norm satisfies the triangle inequalty:
 	$$
-	||x+y|| \leq ||x|| + ||y||
-	$$:
+		||x+y|| \leq ||x|| + ||y||
+	$$
 
 	\vfill
 
@@ -52,7 +52,7 @@
 	Show that the eucidean norm satisfies the reverse triangle inequality:
 
 	$$
-		||x - y|| \geq |~||x|| - ||y||~|
+		||x-y|| \geq |~||x|| - ||y||~|
 	$$
 
 	\vfill
@@ -61,7 +61,7 @@
 	Prove the Cauchy-Schwartz inequality:
 
 	$$
-		||x \cdot y|| = ||x||~||y||
+		||x \cdot y|| \leq ||x||~||y||
 	$$
 
 	\vfill

Advanced/Linear Algebra 101/parts/0 notation.tex

@@ -7,7 +7,7 @@
 	\item $\mathbb{R}^+_0$ is the set of positive real numbers and zero.
 \end{itemize}
 
-Mathematicians are often inconsistent with their notation. Depending on the author, their mood, and the phase of the moon, $\mathbb{R}^+$ may or may not include zero. I will use the definitions above.
+Mathematicians are often inconsistent with their notation. Depending on the author, their mood, and the phase of the moon, $\mathbb{R}^+$ may or may not include zero. We will use the definitions above.
 
 
 \definition{}
@@ -64,12 +64,12 @@ What is $\mathbb{R} \times \mathbb{R}$? \\
 
 \definition{}
 $\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \\
-In english, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\
+In English, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\
 
 \vspace{4mm}
 
 Elements of $\mathbb{R}^2$ look like $(a, b)$, where $a, b \in \mathbb{R}$. \hfill \note{\textit{Note:} $\mathbb{R}^2$ is pronounced \say{arrgh-two.}}
-Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4 a_5)$, where $a_n \in \mathbb{R}$. \\
+Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4, a_5)$, where $a_n \in \mathbb{R}$. \\
 
 $\mathbb{R}^1$ and $\mathbb{R}$ are identical.
 

Advanced/Linear Algebra 101/parts/1 vectors.tex

@@ -57,18 +57,19 @@ Can you develop geometric intuition for their sum and difference?
 \pagebreak
 
 \definition{Euclidean Norm}
-In general, a \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
-Usually, one thinks of a norm as a \say{length metric} on a vector space. \\
+A \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
+Usually, one thinks of a norm as a way of mesuring \say{length} in a vector space. \\
 The norm of a vector $v$ is written $||v||$. \\
 
 \vspace{2mm}
 
-We usually use the \textit{euclidean norm} when we work in $\mathbb{R}^n$. \\
-If $v \in \mathbb{R}^n$, the euclidean norm is defined as follows:
+We usually use the \textit{Euclidean norm} when we work in $\mathbb{R}^n$. \\
+If $v \in \mathbb{R}^n$, the Euclidean norm is defined as follows: \\
+If $v = [v_1, v_2, ..., v_n]$,
 $$
 	||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
 $$
-This is simply an application of the pythagorean theorem.
+This is simply an application of the Pythagorean theorem.
 
 \problem{}
 Compute the euclidean norm of
@@ -78,4 +79,10 @@ Compute the euclidean norm of
 \end{itemize}
 
 \vfill
+
+\problem{}
+Show that $a \cdot a$ is $||a||^2$.
+
+\vfill
+
 \pagebreak

Advanced/Linear Algebra 101/parts/2 dotprod.tex

@@ -11,7 +11,7 @@ The dot product maps two elements of $\mathbb{R}^n$ to one element of $\mathbb{R
 
 	\vspace{2mm}
 
-	It's also worth noting that a function $f$ from $X$ to $Y$ can defined as a subset of $X \times Y$, where for all $x \in X$ there exists a unique $y \in Y$ so that $(x, y) \in f$. Try to make sense of this definition.
+	It's also worth noting that a function $f$ from $X$ to $Y$ can be defined as a subset of $X \times Y$, where for all $x \in X$ there exists a unique $y \in Y$ so that $(x, y) \in f$. Try to make sense of this definition.
 }
 
 $$
@@ -27,9 +27,11 @@ Compute $[2, 3, 4, 1] \cdot [2, 4, 10, 12]$
 Show that the dot product is
 \begin{itemize}
 	\item Commutative
-	\item Distributive
-	\item Homogeneic: $x(a \cdot b) = xa \cdot b = a \cdot xb$
-	\item Positive definite: $a \cdot a \geq 0$, with equality iff $a = 0$
+	\item Distributive $a \cdot (b + c) = a \cdot b + a \cdot c$
+	\item Homogenous: $x(a \cdot b) = xa \cdot b = a \cdot xb$ \\
+	\note{$x \in \mathbb{R}$, and $a, b$ are vectors.}
+	\item Positive definite: $a \cdot a \geq 0$, with equality iff $a = 0$ \\
+	\note{$a \in \mathbb{R}^n$, and $0$ is the zero vector.}
 \end{itemize}
 
 
@@ -40,7 +42,8 @@ Show that the dot product is
 
 
 \problem{}
-Say you have two vectors, $a$ and $b$. Show that $a \cdot b$ = $||a||~||b||\cos(\alpha)$ \\
+Say you have two vectors, $a$ and $b$. Show that $a \cdot b$ = $||a||~||b||\cos(\alpha)$, \\
+where $\alpha$ is the angle between $a$ and $b$. \\
 \hint{What is $c$ in terms of $a$ and $b$?}
 \hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$}
 \hint{The length of $a$ is $||a||$}

Advanced/Linear Algebra 101/parts/3 matrices.tex

@@ -9,7 +9,10 @@ A =
 	4 & 5 & 6
 \end{bmatrix}
 $$
-The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.
+The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix. \\
+\vspace{1mm}
+The order \say{first rows, then columns} is usually consistent in linear algebra. \\
+If you look closely, you may also find it in the next definition.
 
 \definition{}<matvec>
 We can define the product of a matrix $A$ and a vector $v$:
@@ -34,8 +37,8 @@ Note that each element of the resulting $2 \times 1$ matrix is the dot product o
 $$
 Av =
 \begin{bmatrix}
-	\text{---} a_1 \text{---} \\
-	\text{---} a_2 \text{---}
+	\text{---} r_1 \text{---} \\
+	\text{---} r_2 \text{---}
 \end{bmatrix}
 \begin{bmatrix}
 	| \\
@@ -44,20 +47,13 @@ Av =
 \end{bmatrix}
 =
 \begin{bmatrix}
-	r_1v \\
-	r_2v
+	r_1 \cdot v \\
+	r_2 \cdot v
 \end{bmatrix}
 $$
 
 Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix.
 
-
-\problem{}
-Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?
-
-\vfill
-
-
 \problem{}
 Compute the following:
 
@@ -72,6 +68,13 @@ $$
 \end{bmatrix}
 $$
 
+\vfill
+
+
+\problem{}
+Say you multiply a size-$m$ vector $v$ by an $m \times n$ matrix $A$. \\
+What is the size of your result $Av$?
+
 \vfill
 \pagebreak
 
@@ -99,8 +102,8 @@ Note each element of the resulting matrix is dot product of a row of $A$ and a c
 $$
 AB =
 \begin{bmatrix}
-	\text{---} a_1 \text{---} \\
-	\text{---} a_2 \text{---}
+	\text{---} r_1 \text{---} \\
+	\text{---} r_2 \text{---}
 \end{bmatrix}
 \begin{bmatrix}
 	|	& | \\
@@ -109,8 +112,8 @@ AB =
 \end{bmatrix}
 =
 \begin{bmatrix}
-	r_1v_1 & r_1v_2 \\
-	r_2v_1 & r_2v_2 \\
+	r_1 \cdot v_1 & r_1 \cdot v_2 \\
+	r_2 \cdot v_1 & r_2 \cdot v_2 \\
 \end{bmatrix}
 $$
 
@@ -263,7 +266,7 @@ Vertical arrays don't look good in horizontal text.
 \problem{}
 Consider the vectors $a = [1, 4, 3]^T$ and $b = [9, 1, 4]^T$ \\
 \begin{itemize}
-	\item Compute the dot product $ab$.
+	\item Compute the dot product $a \cdot b$.
 	\item Can you redefine the dot product using matrix multiplication?
 \end{itemize}
 \note{As you may have noticed, a vector is a special case of a matrix.}