18 03 4 at ESG Spring 2003 Comments on Hirsch Smale Chapter 3 The Chapter and Section titles are a bit misleading Look at the title of Chapter 3 4 and perhaps ask what this is doing in this chapter instead of the next in which case perhaps ask what Chapter 4 2 is doing there as well All in good time This editor s opinion is that Chapter 3 serves the dual purposes of introducing or reviewing the basics of linear algebra and getting underway in actually solving some differential equations putting that linear algebra to use For those not familiar with linear algebra an analogy would be teaching someone to swim by throwing such a person into a cold lake If this is indeed the case keep in mind that this editor is also a trained lifeguard In page order then 1 Basic Linear Algebra We emphasize that for many readers this section should be used only as a reference or a review Note how wonderfully qualified this statment is Fortunately it s true We will be using this section primarily as a reference so expect references to this section often Rather than absorb all of this at once we ll look at the material as needed So perhaps keep these comments handy for well reference Page 30 near the middle With the additional axiom of finite dimensionality abstract vector spaces could be used in place of Rn throughout most of this book Indeed we will reserve the option of doing so and in fact we ll even use spaces that are not finite dimensional function spaces such as the space H 1 a b the set of square integrable functions on a b should be considered subject to these VS1 and VS2 rules Starting around here the authors no longer qualify the term operator and unless explictily mentioned otherwise all operators should be assumed to be linear operators Page 33 Those who have used Apostol s Volume II should recognize that the terminology for inverses of operators is a bit different in at least two respects First Apostol distinguishes between right and left inverses a distinction we will not need to make and so this terminology will not be used this term Next the symbol A 1 has two meanings on one page and the meaning must be inferred from the usage The first usage at the top of the page is for the inverse of a matrix 1 assuming the matrix is invertable At the bottom of the page A is a linear map which could be represented as a matrix and A 1 0 is iterpreted as all elements which are mapped to 0 For concrete examples of this second usage consider A R3 R1 defined by A x c x where c R3 is a contstant vector and the dot product is the traditional one Then A 1 0 is the set of vectors in R3 which are mapped to 0 in this case all vectors perpendicular to c which we can and should interpret as a plane containing the origin As another example consider A H 1 a b H 1 a b defined by A f t df dt Verifying that this is a linear map is quite direct and A 1 0 is the set of all functions whose derivative is 0 all functions constant on a b By the way in the above usage 0 is not the scalar 0 but the function which is identically zero on a b There is a difference but writing 0 0 t which I just did looks foolish which I just was Two other things while we re here The text says Henceforth we shall use the term vector space to mean subspace of a Cartesian space I don t believe this for a minute but let s wait and see what happens In our class we will not be making this restriction Also the kernel of a linear map is often called the null space for reasons which might be obvious You are likely to encounter this alternate term in any of several venues especially linear algebra Apostol uses this terminology Page 34 The definition of a basis repeated here is A basis of F is an ordered set of vectors in F that is independent and which spans F It should be noted that the ordered is important For instance in R3 one basis would be i j k and another would be k j i Despite appearances these are not considered the same basis To cut to the chase we know that the first is a right handed coordinate system while the other is left handed This does matter and if you recall Jacobians from 18 02 or its equivalent you ll recall that when we change coordinate systems changing from right to left handed systems introduces a minus sign Page 35 The italicized statement Solving the system means finding a basis for Ker A is italicized for a reason Finding a Kernel is more general than solving a system Page 36 C Changes of bases and coordinates It s important to recognize what s going on in the map E Rn 2 The vector z E is an element of E which we are calling a vector The map takes this element into an n tuple of real numbers The point is to distinguish between a vector and the set of numbers that describe that vector To appreciate the distinction take your favorite vector gravitational acceleration g If you choose your axes such that k is down then you would say g 0 0 g where g g However if you do your work on a blackboard and use j as up you would say g 0 g 0 You haven t changed the vector you re not allowed to but you ve changed the coordinates that describe your favorite vector In the middle of the page we re told that We thus have three equivalent concepts a basis of E a coordinate system on E and an isomorphism E Rn This is quite an important concept and you have to convince yourself of its validity Page 37 near bottom it s stated that AB t B t At for any n n matrices A B This is true and is stated without proof I didn t see it mentioned earlier It s not hard to show by considering the elements of the product AB as being the respective dot products of the rows of A and the columns of B Page 38 D Operator bases and matrices and shouldn t that be Operators Here the notation is a bit different Recall that the symbol xi is used for both one of the elements of an ordered list and the linear function defined by xi ej ij on Page 36 That is in Equation 9 xi T is the product of two linear operators as mentioned in the last paragraph on this page In passing the authors mention that Equation 9 looks very pretty when placed next to 7 a phrase not often found in texts of this nature …
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