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Unformatted text preview: MATH270 SUMMARY READ WITH SUSPICION APRIL 1, 2010 The following represents a snapshot of the course as I see it. The aim is to isolate the important conceptual developments and give some sense of how it fits together. This is of course highly subjective and almost certainly incomplete. Let me know if you wish to add anything! (1) Complex Numbers We need these for any intelligent discussion of eigenvalues and eigenvectors, be- cause of the fundamental theorem of algebra. But it doesn’t hurt to see them in action earlier. You should be able to solve equations like z 5 = 3 and find all 5 solutions. Don’t forget the polar form is very useful if you have to multiply and divide complex numbers! (2) Elimination We spent a long time on this. It and its improvements are among the most useful algorithms. But actually performing algorithms should be left to computers. Our job is to study, understand and create them! Here, we encoded elimination as the matrix factorisation PA = LU . You should know all about this by now! (3) Fields and Vector Spaces These form the basis (I am so funny!) of all linear algebra. The only fields we’ve really cared about are R and C , and they haven’t played a big role in the course, ex- cept when I’ve tried to convince you that C is infinitely better. Here, we introduced subspaces, linear independence, spanning sets, bases and dimensions. A good un- derstanding of these concepts with low-dimensional examples in mind can only be a good thing. (4) Fundamental Subspaces of a Matrix The first application of subspace technology was to introduce the fundamental sub- spaces of a matrix A . You should know what they are, how to find them, and most importantly, what they are good for: They control (among other things) the existence and uniqueness of solutions to Ax = b . Moreover, the algorithms we use to compute them are also invaluable in finding a basis for an arbitrary subspace and testing if vectors belong to a given subspace! These algorithms are the workhorses of linear algebra. We defined things like rank and nullity here, not out of any sense of malice, but because naming something that comes up again and again is just plain efficient. Know the theorems please, especially any with names like “fundamental”. 1 2 APRIL 1, 2010 (5) Linear Transformations This is an important digression. In the real world, you’re often not handed a matrix on a plate. You just know that something is happening and maybe you wish to model it in the simplest way. This usually leads to a linear transformation. Once you choose a basis, you end up with a matrix to which you can apply your vast arsenal of tricks. This process is what we called a representing matrix. Its importance in the physical sciences is difficult to overstate. We also studied the effect on the representing matrix upon changing the basis. This led to a formula eerily reminiscent of diagonalising a matrix. Why? Because diagonalisation is nothing but changing to a basis consistingmatrix....
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