Improve lake model

[jstac]

@HengchengZhang , could you please review the lecture lake_model.md as follows:

There are references to the Perron-Frobenius theorem. They should be linked to our presentation of that theorem in eigen_II.md.

The exercise and solution environments need to be set up properly.

Tidy up any messy equations --- used aligned if necessary.

Spell check.

Break up into subsections. Start the lecture with ## Outline. Then perhaps ## The Model followed by ## Dynamics.

"colsum" should not be italic.

Use proper inner product symbols.

[HengchengZhang]

Hi @jstac, when fixing the messy equations I noticed that the way we use matrices and vectors is quite different in different lectures, like in Markov Chains.

Shall we also be uniform at this?

Also a note to myself:

• Reference eigen_II.md
• Set up exercise environment
• Tidy up equations
• Spell check
• Break up into subsctions
• Non-italic colcum
• Proper inner product

[jstac]

Thanks @HengchengZhang .

What do you mean by "the way we use"? Can you give an example?

Do you mean the notation $P(x,y)$ instead of $P_{ij}$?

(It's a bit irregular but helpful in the case of Markov chains.)

[HengchengZhang]

Thanks John, it's the use of parentheses(using bmatrix) and brackets(using array).

[jstac]

Ah, you mean that we use pmatrix instead of bmatrix? Good catch.

(Please be ask specific as possible with your comments so I can figure out what you mean quickly.)

The style guide says we should use bmatrix so I guess we'd better follow it (even though I prefer pmatrix...)