~~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 =====

  * Some (useful) ways of taking two matrices and making a new matrix.
  * $1\times 1$ matrices are the same as numbers....
  * ....we'll define some operations on matrices which generalise operations on numbers, like addition and multiplication

==== Matrix addition and subtraction ====

{{page>matrix addition}}

==== Examples ====
  * $ \left[\begin{smallmatrix}1&2&-2\\3&0&5\end{smallmatrix}\right]+\left[\begin{smallmatrix}-2&2&0\\1&1&1\end{smallmatrix}\right]=\left[\begin{smallmatrix}-1&4&-2\\4&1&6\end{smallmatrix}\right].$
  * $\left[\begin{smallmatrix}1&2&-2\\3&0&5\end{smallmatrix}\right]+\left[\begin{smallmatrix}-2&2\\1&1\end{smallmatrix}\right]\text{ is undefined.}$

==== Remarks ====

  - For any matrices $A$ and $B$ with the same size: $A+B=B+A$. We say that matrix addition is //commutative//.
  - For any matrices $A$, $B$ and $C$ with the same size: $(A+B)+C=A+(B+C)$. We say that matrix addition is //associative//.

==== The zero matrix ====

{{page>zero matrix}}

==== Exercise ====

Show that if $A$ is any $n\times m$ matrix, then 
\[ 0_{n\times m}+A=A=A+0_{n\times m}.\]

  * Remember that when checking that matrices are equal, you have to check that they have the same size, and that all the entries are the same.

==== Definition of matrix subtraction ====

{{page>matrix subtraction}}

==== Examples ====
  * $ \left[\begin{smallmatrix}1&2&-2\\3&0&5\end{smallmatrix}\right]-\left[\begin{smallmatrix}-2&2&0\\1&1&1\end{smallmatrix}\right]=\left[\begin{smallmatrix}3&0&-2\\2&-1&4\end{smallmatrix}\right].$
  * $ \left[\begin{smallmatrix}1&2&-2\\3&0&5\end{smallmatrix}\right]-\left[\begin{smallmatrix}-2&2\\1&1\end{smallmatrix}\right]\text{ is undefined.}$

==== Scalars ====

  * a **scalar** is just a fancy name for a number
    *  (in this course: a real number)
  * Why use this strange-looking name?
    * numbers are often used for scaling things up or down
    * e.g. the scalar 3 is used to scale things up by a factor of 3 (by multiplying by 3).

==== Scalar multiplication of matrices ====
{{page>scalar multiplication of matrices}}

==== Example ====


  * If $A=\left[\begin{smallmatrix}1&0&-3\\3&-4&1\end{smallmatrix}\right]$, then $3A=\left[\begin{smallmatrix}3&0&-9\\9&-12&3\end{smallmatrix}\right]$. 
  * In other words, $ 3\left[\begin{smallmatrix}1&0&-3\\3&-4&1\end{smallmatrix}\right]=\left[\begin{smallmatrix}3&0&-9\\9&-12&3\end{smallmatrix}\right].$

==== The negative of a matrix ====
{{page>matrix negation}}

==== Exercise ====
Prove that $A-B=A+(-B)$ for any matrices $A$ and $B$ of the same size.

==== Row-column multiplication ====

  * 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.
  * The **row-column product** of $a$ and $b$ is defined by \[\!\!\!\!\!\!\!\!\!\!ab=[\begin{smallmatrix}a_1&a_2&\dots&a_n\end{smallmatrix}]\left[\begin{smallmatrix}b_1\\b_2\\\vdots\\b_n\end{smallmatrix}\right]=a_1b_1+a_2b_2+\dots+a_nb_n.\]
  * Sometimes we write $a\cdot b$ instead of $ab$.
  * If $a$ and $b$ have a different number of entries, $ab$ is **undefined**.

==== Examples ====

  * $\left[\begin{smallmatrix}1&2\end{smallmatrix}\right]\left[\begin{smallmatrix}3\\-1\end{smallmatrix}\right]=1\cdot 3+2\cdot(-1)=3+(-2)=1$.
  * $\left[\begin{smallmatrix}1&2&7\end{smallmatrix}\right]\left[\begin{smallmatrix}3\\-1\end{smallmatrix}\right]$ is not defined.
  * $\left[\begin{smallmatrix}2&3&5\end{smallmatrix}\right]\left[\begin{smallmatrix}x\\y\\z\end{smallmatrix}\right]=2x+3y+5z$.
  * 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 (not just rows and columns)
  * This will generalise row-column multiplication
  * We build $AB$ from all possible row-column products. 
  * For example: \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \def\r{\left[\begin{smallmatrix}1&0&5\end{smallmatrix}\right]}\def\rr{\left[\begin{smallmatrix}2&-1&3\end{smallmatrix}\right]}\left[\begin{smallmatrix}1&0&5\\2&-1&3\end{smallmatrix}\right]\left[\begin{smallmatrix} 1&2\\3&4\\5&6\end{smallmatrix}\right]\def\s{\left[\begin{smallmatrix}1\\3\\5\end{smallmatrix}\right]}\def\ss{\left[\begin{smallmatrix}2\\4\\6\end{smallmatrix}\right]}=\left[\begin{smallmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{smallmatrix}\right]=\left[\begin{smallmatrix}26&32\\14&18\end{smallmatrix}\right].\]