Differences

This shows you the differences between two versions of the page.

--- lecture_8_slides [2016/02/16 17:48] – rupert
+++ lecture_8_slides [2017/02/16 08:56] (current) – [Commuting matrices III] rupert
@@ Line 1: / Line 1: @@
 ~~REVEAL~~
-==== Matrix multiplication ====
+==== Row-column & matrix multiplication ====
+  * The **row-column product** of $a$ and $b$ is defined by \[\!\!\!\!\!\!\!\!\!\!ab=[\begin{smallmatrix}a_1&a_2&\dots&a_n\end{smallmatrix}]\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]=a_1b_1+a_2b_2+\dots+a_nb_n.\]
+  * $AB=$ matrix of all "row-of-$A$ times col-of-$B$" products
+  * \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]
+==== Matrix multiplication: the definition ====
   * Let $A,B$ be matrices, with sizes
@@ Line 25: / Line 32: @@
 ==== Example 2 ====
-If $A=\mat{1&2\\3&4\\5&6}$, $B=\mat{2&1&1\\1&2&0\\1&0&2\\1&0&2\\2&2&1}$ and $C=\mat{1&3&0&7\\0&4&6&8}$,
+If $A=\mat{1&2\\3&4\\5&6}$, $B=\mat{2&1&1\\1&2&0\\1&0&2\\2&2&1}$ and $C=\mat{1&3&0&7\\0&4&6&8}$,
   * $A$ is $3\times 2$, $B$ is $4\times 3$ and $C$ is $2\times 4$, so
   * $AB$, $CA$ and $BC$ don't exist (undefined);
@@ Line 62: / Line 69: @@
 {{page>commute}}
-  * Because it's not true in general that $AB=BA$, we say that **matrix multiplication is not commutative**.
+  * Because it's true that $AB=BA$ for every choice of matrices $A$ and $B$, we say that **matrix multiplication is not commutative**.
 ==== Commuting matrices II ====
-  * What can we say about commuting matrices?
+  * What can we say about a pair of commuting matrices?
   * Suppose $AB=BA$ and think about sizes.
     * $A$: $n\times m$
@@ Line 77: / Line 84: @@
   * If $A$ and $B$ commute, they must be square matrices of the same size.
-  * **Some** square matrices $A$ and $B$ of the same size commute...
+  * **Some** pairs of square matrices $A$ and $B$ of the same size do commute...
   * ....but not all!
   * See examples above.