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--- lecture_7_slides [2016/02/10 14:52] – rupert
+++ lecture_7_slides [2017/02/13 19:32] (current) – rupert
@@ Line 1: / Line 1: @@
 ~~REVEAL~~
+====== Chapter 2: The algebra of matrices ======
+==== ====
+{{page>matrix}}
+{{page>(i,j) entry}}
+==== Examples ====
+  * $B=\begin{bmatrix} 99&3&5\\7&-20&14\end{bmatrix}$ is a $2\times 3$ matrix
+    * the $(1,1)$ entry of $B$ is $b_{11}=99$
+    * the $(1,3)$ entry of $B$ is $b_{13}=5$
+    * the $(2,1)$ entry of $B$ is $b_{21}=7$
+    * etc.
+  * $(3,2)$ entry of $B$?
+    * undefined!
+==== Examples =====
+  * $\left[\begin{smallmatrix}3\\2\\4\\0\\-1\end{smallmatrix}\right]$ is a $5\times 1$ matrix.
+    * A **column vector**: matrix with just one column.
+  * $\begin{bmatrix}3&2&4&0&-1\end{bmatrix}$ is a $1\times 5$ matrix.
+    * A **row vector**: matrix with just one row.
+  * These have the same entries, but a different "shape", or "size".
+  * They are different matrices (not equal).
+==== Size of a matrix ====
+{{page>same size}}
+==== Equality of matrices ====
+{{page>equal matrices}}
+==== Examples ====
+  * $\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}\ne \begin{bmatrix}3&2&4&0&-1\end{bmatrix}$, since these matrices have different sizes: the first is $5\times 1$ but the second is $1\times 5$.
+==== ====
+  * $\begin{bmatrix}1\\2\end{bmatrix}\ne\begin{bmatrix}1 &0\\2&0\end{bmatrix}$
+    * not the same size.
+  * $\begin{bmatrix}1&0\\0&1\end{bmatrix}\ne \begin{bmatrix}1&0\\1&1\end{bmatrix}$
+    * same size but the $(2,1)$ entries are different.
+==== ====
+  * If $\begin{bmatrix}3x&7y+2\\8z-3&w^2\end{bmatrix}=\begin{bmatrix}1&2z\\\sqrt2&9\end{bmatrix}$ then we know that all the corresponding entries are equal
+  * We get four equations:\begin{align*}3x&=1\\7y+2&=2z\\8z-3&=\sqrt2\\w^2&=9\end{align*}
 ===== Operations on matrices =====
@@ Line 64: / Line 109: @@
 ==== Row-column multiplication ====
-  * Let $a=[a_1~a_2~\dots~a_n\end{bmatrix}]$ be a $1\times n$ row vector
+  * Let $a=[\begin{smallmatrix}a_1&a_2&\dots&a_n\end{smallmatrix}]$ be a $1\times n$ row vector and $b=\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]$, an $n\times 1$ column vector.
-  * and $b=\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]$, an $n\times 1$ column vector.
+  * The **row-column product** of $a$ and $b$ is defined by \[\!\!\!\!\!\!\!\!\!\!ab=[\begin{smallmatrix}a_1&a_2&\dots&a_n\end{smallmatrix}]\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]=a_1b_1+a_2b_2+\dots+a_nb_n.\]
-  * The **row-column product** $ab$ of $a$ and $b$ is defined by \[[a_1~a_2~\dots~a_n]\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]=a_1b_1+a_2b_2+\dots+a_nb_n.\]
   * Sometimes we write $a\cdot b$ instead of $ab$.
   * If $a$ and $b$ have a different number of entries, $ab$ is **undefined**.
@@ Line 75: / Line 119: @@
   * $\left[\begin{smallmatrix}1&2&7\end{smallmatrix}\right]\left[\begin{smallmatrix}3\\-1\end{smallmatrix}\right]$ is not defined.
   * $\left[\begin{smallmatrix}2&3&5\end{smallmatrix}\right]\left[\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right]=2x+3y+5z$.
-  * Generalising the previous example: if $a=\left[\begin{smallmatrix}a_1&a_2&\dots&a_m\end{smallmatrix}\right]$ and $x=\left[\begin{smallmatrix}x_1\\x_2\\\vdots\\x_m\end{smallmatrix}\right]$, then $ax=a_1x_1+a_2x_2+\dots+a_mx_m$. So we can write any [[linear equation]] $a_1x_1+a_2x_2+\dots+a_mx_m=b$ as a shorter matrix equation: $ax=b$.
+  * If $a=\left[\begin{smallmatrix}a_1&a_2&\dots&a_m\end{smallmatrix}\right]$ and $x=\left[\begin{smallmatrix}x_1\\x_2\\\vdots\\x_m\end{smallmatrix}\right]$, then $ax=a_1x_1+a_2x_2+\dots+a_mx_m$.
+    * So we can write any [[linear equation]] as a shorter matrix equation: $ax=b$.
 ==== Matrix multiplication ====
-  * We want to define $AB$ where $A$ and $B$ are matrices of "compatible" sizes
+  * We want to define $AB$ where $A$ and $B$ are matrices of "compatible" sizes (not just rows and columns)
   * This will generalise row-column multiplication
-  * Idea is to build the new matrix $AB$ from all possible row-column products
+  * We build $AB$ from all possible row-column products.
-  * Here's an example:\[ \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]
+  * For example: \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]