Strang teaches it backwards from how I was taught. Instead of starting with determinants and computational recipes, he starts with the space spanned by a matrix's columns and with the idea that multiplying by a matrix transforms space. With that in hand, everything else stops being a recipe.
What I took from it
The geometric reading. A matrix is not a table of numbers, it is a transformation, and the useful questions about it are what its columns span and how much genuinely independent information sits inside. That has a direct consequence in my work: when someone shows me a dashboard with twenty indicators that all move together, they do not have twenty indicators, they have three wearing twenty costumes. It is the same phenomenon as the rank of a matrix.
The second thing is understanding what a meaning vector is, which is the piece holding up much of what gets sold today as artificial intelligence. Knowing that similarity between two texts is being measured as an angle between two arrows changes what I promise: a small angle means similar in how words occur, not true and not relevant.
And a third, less technical: least squares is a projection. The regression everyone uses to forecast sales predicts nothing; it finds the nearest point inside what the model is able to represent. Put that way, it is obvious why it fails when what is coming does not resemble what already happened.
What I left out
The exercises and the numerical detail. And one honest observation: this is a course book, meant to accompany lectures. The recorded lectures the author left public carry these ideas better than the pages do, and anyone choosing between the two should watch the lectures.