10. The Four Fundamental Subspaces
OK, here is lecture ten in linear algebra. Two important things to do in this lecture. One is to correct an error from lecture nine. So the blackboard with that awful error is still with us. And the second, the big thing to do is to tell you about the four subspaces that come with a matrix. We've seen two subspaces, the column space and the null space. There's two to go. First of all, and this is a great way to OK. recap and correct the previous lecture -- so you remember I was just doing R^3. I couldn't have taken a simpler example than R^3. And I wrote down the standard basis. That's the standard basis. The basis -- the obvious basis for the whole three dimensional space. And then I wanted to make the point that there was nothing special, nothing about that basis that another basis couldn't have. It could have linear independence, it could span a space. There's lots of other bases. So I started with these vectors, one one two and two two five, an...