Applied edits

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.}