Reading a Matrix This is a softer question, but I'm having trouble keeping straight all of the information that a matrix provides you with straight in my head. All I know is that the rows correspond with the codomain and the columns correspond with the domain. Is there a pictural representation associating a matrix with it's inputs and it's outputs? 
 A: Finding pictures for things -- always a praiseworthy pursuit.
There are a bunch of ways to think about matrices, so there's no one picture. But here's a picture, of a matrix acting on a vector:

This picture is from BetterExplained. Click on the link, there's a whole article about this.
The picture encourages you to imagine a matrix, for example:
$$
\begin{bmatrix}
1 & -1 & 0\\
0 & 0 & 2 \\
4 & 0 & 1 \\
\end{bmatrix}
$$
as a list of dot products to take: it says, when you feed me a vector $x = (a, b, c$), I spit out a new vector, the first entry of which is $x \cdot (\text{my first row})$, the second being $x \cdot (\text{my second row})$, and so on.
For example, this matrix:
$$
\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}
$$
when you feed it a 2-element vector, spits out that vector but with the elements flipped. Can you see why?
This is a spreadsheet-y, engineering-oriented viewpoint on matrices -- viewing them as linear transformations of $\mathbb{R}^n$ leads to a different and equally helpful mental picture.
I usually find BetterExplained thought-provoking, even if the mental picture he presents doesn't end up being my One True intuition.
