@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.