(Or how to see the 10,000 dimensional space.)
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.
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.
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