\\section{Matrices as Arrays}

The simplest way we think of matrices is as arrays of numbers, with an associated way to \'multiply\' them, yeilding new such arrays:

$$[\\begin{array}
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square
\\end{array}]
\\CircleTimes
[\\begin{array}
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square
\\end{array}]
=
[\\begin{array}
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square
\\end{array}]
$$
This \'multiplication\' can also \'apply\' a matrix $M$ to a vector $\\vec{V}$:
$$M \\CircleTimes \\vec{V} = 
[\\begin{array}
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square
\\end{array}]
\\CircleTimes
(\\begin{array}
\\Square\\\\
\\Square\\\\
\\Square
\\end{array})
=
(\\begin{array}
\\Square\\\\
\\Square\\\\
\\Square
\\end{array})$$
Provide an \'inner product\' mapping two vectors to a scalar:
$$(\\begin{array}
\\Square&\\Square&\\Square
\\end{array})
\\CircleTimes
[\\begin{array}
\\Square\\\\
\\Square\\\\
\\Square
\\end{array}]
=[\\Square]
$$
Or even an \\it{Outer Product}, seldom discussed but very useful and interesting, which takes two vectors and returns a matrix:
$$

[\\begin{array}
\\Square\\\\
\\Square\\\\
\\Square
\\end{array}]
\\CircleTimes
(\\begin{array}
\\Square&\\Square&\\Square
\\end{array})
=
[\\begin{array}
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square\\\\
\\Square&\\Square&\\Square
\\end{array}]
$$