Background on linear algebra #21
Description
Friday's lecture will be on the orthogonal projection theorem and its applications. To follow the lecture you will require some background in linear algebra. Because of this I've prepared some background slides. See the lecture 7 directory.
In the lecture I will assume you know what the span of a collection of vectors is, what linear independence means, what a linear subspace is and what a basis of a linear subspace means.
I've also put in some material on solving linear systems and nonsingularity, towards the end of the slides. That material is not essential for Friday's lecture. But it is fundamental topic. It took me a while to realize that the best way to understand linear systems of equations is to think about the corresponding linear map and ask whether it's a bijection. Then we can exploit the fact for linear maps from R^n to R^n, the property of being a bijection is equivalent to being either onto or one-to-one. Once we know this we can go back to the matrix problem and ask what properties we require on the matrix so that the corresponding linear map is onto or one-to-one.
Unfortunately we won't have time to cover this material in class. Please let me know if you spot typos.