Usually every mathematical area has the most important theorem, Fundamental theorem. In linear algebra, it is called Fundamental theorem of linear algebra. This is about the relationships between four subspaces: Column space, row space, null space, and left null space. But, I don't recall I heard them in my mathematics course in my university. Did I miss that? If I miss that, that is one of the most significant misses. Because I did not know the column space, I had so hard time to figure out high dimensional spaces. I know the row space (though I did not know it has a name), but this doesn't give me a high dimensional space image. I even can not imagine four dimensional space without the column space. But now, I know the column space and it is easy to imagine what is a 10,000 dimensional space. If you are interested in, please see my slides. Or you have more time, read the Gilbert Strang's Introduction to Linear Algebra.
3 days ago