MA-305 Forum


Please submit all question to the TA, John Haws (send him e-mail). All postings will remain anonymous.
  1. Miscellaneous
  2. Homework Misc.
  3. Homework 1
  4. Homework 2
  5. Homework 3
  6. Homework 4
  7. Homework 5
  8. Homework 6
  9. Maple
    1. refpkg
    2. lsqpkg
    3. misc
  10. Lectures on the Internet
    1. Audio Files
    2. Slides

  1. Miscellaneous Questions


    Question: I don't understand this business about groups. Can you explain a little about this, and maybe give some examples?
    Answer: Groups are formally defined as follows: any set S together with a binary operation "°" is called a group if the following properties hold:
    1. There exists in S an identity element e relative to the binary operation "°":
      a ° e = e ° a = a, for all a in S.
    2. The binary operation is associative:
      a ° (b ° c) = (a ° b) ° c, for all a, b, c in S.
    3. Every element a in S has an (unique) inverse element a-1 in S:
      a ° a-1 = a-1 ° a = e, for all a in S.
    Examples: (1)Consider the integers as a set. If your binary operation is "+", then we have a group, because we have an identity element, 0, and for each integer n, an inverse, -n. Using the notation we used in class, we would write this (Z; +, 0, -).
    However, if the binary operation is multiplication, we don't have a group, because we have trouble finding inverses. For example, the inverse of 2, under multiplication, would be ½, which is not an integer.
    (2)If we consider our set all nxn invertible matrices, with matrix multiplication as our operation, we have a group. The identity would be In, and the multiplicative inverse of each matrix exists. Using the notation we used in class, we would write this (GLn(R); °, In, -1).
    Now, this set is not a group under addition, because it is not closed (necessary under the definition of a binary operation): if we add two invertible matrices, the sum may not be invertible!

  2. Homework Miscellaneous


    Question: I want to submit my homework as a Maple worksheet. Should it be formatted in a special way? And is there a preferred naming convention?
    Answer: Send an email to John Haws (jchaws@eos.ncsu.edu), and attach your file to the e-mail. You can send a file saved as Maple Text (the preferred format), or a file saved as a Maple Worksheet (.mws) or a simple ASCII file.
    An appropriate name would be hw#_username.mws, where # represents the assignment number, and username represents your login id. For example, I would submit Homework 2 as hw2_jchaws.mws, if I were submitting it as a maple worksheet.
    Question: Can I submit my homework in any other format?
    Answer: Yes. You may also type your homework up as an ASCII text file. Attach it to an email, or include it in the body of a message. Of course, you can always submit it in-person in class.
    Question: Do you accept late homework?
    Answer: All homeworks must be submitted on time. The following penalities are given for (unexcused) late submissions:
    • up to 1 day late: 20% reduction
    • up to 2 days late: 50% reduction
    • after 2 days: no credit (= 100% reduction)

  3. Homework 1


    Question: Should I use solve or mysolve to do the homework?
    Answer: You can use whatever you want. mysolve provides more detailed and instructive input (with printlevel := 3), but the choice is yours. You can even do it by hand if you want.

  4. Homework 2


    Question: In problem 2, what does I2 represent?
    Answer: In general, In denotes the nxn identity matrix. In this case, we have
    I2 = 
    [ 1 0 ]
    [ 0 1 ]
    Question: In question 1, what exactly do you mean when you say "describe the geometric properties of the solution set"?
    Answer: Here's a hint: think of the case of n equations in 3 unknowns. The solution set will lie in R3: it will be a point, a line, a plane, or the whole space, etc. Can you say something specific about the geometric properties, then, for homogeneous systems, in general? Again, if you need help, consider some examples in 'easy' spaces, such as R3 or R2.
    Question: In question 4, should the matrix be something like the one we discussed in class? If so, where the heck did it come from?
    Answer: For the standard Fibonacci numbers, we found that the matrix 
    [ 0 1 ]
    [ 1 1 ]

    did the trick. Here's the idea: when we update, our 'new' fn-1 (the first entry in the 'new' vector) is the 'old' fn. The top row, 
    [ 0 1 ]
    just takes the second element from the 'old' vector. Now, our 'new' fn is just the sum of the 'old' two elements. The second row, 
    [ 1 1 ]
    adds up all the elements in the vector it's multiplying; that's exactly what we wanted.
    Now, for our problem, we are going to have to consider more states. Try considering fn-2, etc., and try to find corresponding matrices that do the trick.

  5. Homework 3


    Question: Question 2 is a huge mess; it's taking me for ever to work through it. Am I doing something wrong?
    Answer: Well, you can work through it by hand, but that will be tedious and painful. I suggest using maple; you may need to simplify your answer using a command like map(simplify, X);, or something similar.
    Question: I don't understand this question. What is this matrix T?
    Answer: Recall that T is the transforming matrix, actually just the product of elementary matrices that transform A into a row-echelon form. (It is interesting to note that if U were to be reduced row-echelon form, T would actually be A-1.)
    Question: In question 5, I'm having some trouble. Am I missing something?
    Answer: Here's a hint: Use the fact that for any matrices A and B (or L and U),
          det(AB)=det(A)det(B).
    Also, those are one's on the diagonal of the left matrix L.

  6. Homework 4


    Question: When we talk about vector spaces, what are these 18 axioms I keep hearing about, and how is that related to verifying that a given set is a vector space?
    Answer: Depending on how you phrase the definition of a vector space, you can write it many different ways. If you in detail lay out each point separately, you should find 18 total axioms to be satisfied. But the definition can be stated more concisely. Prof. Kaltofen boils it down to four points (see slide 9, lecture 13):
         For a give vector space V,
    1. (V; +, 0, -) forms an abelian group.
    2. The scalars distribute over the sum of two vectors, and the sum of two scalars distribute over a vector.
    3. Scalar multiplication is associative
    4. There exists, with resp. to the scalar mult., an identity scalar (i.e. 1R)
    To show a given set with given operations is a vector space, it suffices to demonstrate that the set and operations satisfy these four points.
    So, for example, suppose I am given V = {all 1x2 real valued vectors} as my set, with normal vector addition and scalar multiplication.
    1. (V; +, 0, -) forms an abelian group, for let x, y, and z be elements of V, say (x1, x2), (y1, y2), and (z1, z2) respectively. Then
      1. ((x1, x2) + (y1, y2)) + (z1, z2) = (x1 + y1, x2+ y2) + (z1, z2) = (x1 + y1 + z1, x2+ y2 + z2) = (x1, x2) + (y1 + z1, y2 + z2) = (x1, x2) + ((y1, y2) + (z1, z2))
      2. (x1, x2) + (0, 0) = (0, 0) + (x1, x2) = (x1,x2)
      3. (x1, x2) + (-x1, -x2) = (x1 + -x1, x2 + -x2) = (0, 0)
      4. (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2) = (y1 + x1, y2 + x2) = (y1, y2) + (x1, x2)
    2. The scalars distribute over the sum of two vectors, and the sum of two scalars distribute over a vector, for let a, b be real numbers. Then
      a((x1, x2) + (y1, y2)) = a(x1 + y1, x2 + y2) = (ax1 + ay1, ax2 + ay2) = (ax1, ax2) + (ay1, ay2) = a(x1, x2) + a(y1, y2)
      and
      (a+b)(x1, x2) = ((a+b)x1, (a+b)x2) = (ax1 + bx1, ax2 + bx2) = (ax1, ax2) + (bx1, bx2) = a(x1, x2) + b(x1, x2)
    3. (ab)(x1, x2) = (abx1, abx2) = a(bx1, bx2) = a(b(x1, x2))
    4. 1(x1, x2) = (1x1, 1x2) = (x1, x2)
    And hence V is a vector space.
    Question: How do you test for linear independence and span with vectors that have variables in them (e.g. -- problem 6, part c)?
    Answer: One way is to use the formal definition for linear independence: the n vectors x1, x2, ..., xn are lin. independent if and only if the following condition holds:
         a1x1+ a2x2+ ...+ anxn = 0 => a1=a2=...=an=0.

    So for this problem, just put some coefficients in front of the vectors, add them up and set the sum equal to zero, and see if that forces the coefficients to be zero.
    A more difficult part is checking if they span the vector space. For that, you also need to verify that they span the space, i.e. that any element of the space can be represented as a linear combination of the elements in the basis. One way to do this is to pick a generic element (a + bx + cx2 + dx3, where a, b, c, d are rationals, would be a good choice) of the space, and see if you can find coefficients for the given vectors in the basis so that you can add them up and get the generic element.


  7. Homework 5


    Question:How do I find a basis for the row and column space of a matrix?
    Answer:Let's say we have a matrix A and its reduced row-echelon form, say Aref. The the columns of A corresponding to the columns of Aref containing pivot points span the col. space of A; the non-zero rows of Aref span the row space. (Make sure they're independent. Will they always be?)
    Question:When we're asked to find the projection of one vector onto another, do we find that alpha from lecture 19, slide 8?
    Answer:Actually, you need to find alpha*y.

  8. Homework 6


    Question:How do I find a basis for the Orthogonal Complement to a matrix?
    Answer:We are considering the Orthogonal Complementto the space spanned by the columns of the matrix -- i.e. the Column Space. The Orthogonal Complement to that space would be the set of vectors that are orthogonal to the space spanned by the columns. So we really need just to find a basis for the nullspace of the transpose of the matrix.
    Question:In problem #3, what is this matrix A that I am asked to find, and how do I find it?
    Answer:Slide 12, Lecture 27, begins an explanation and example for this. This matrix depends on which basis you choose, but let's assume we are given a basis. To find the matrix, just perform the transform on the basis elements, and write the resulting vectors as columns in a matrix. The matrix you get is the matrix asked for.
    Now for pt. c, instead of considering T:M2x2 -> M2x2, consider T:R4 -> R4 where
    T( [
    [
    [
    [    		 w
    x
    y
    z    		 ]
    ]
    ]
    ]    		 ) = [
    [
    [
    [    		 w+z
    0
    0
    x+y    		 ]
    ]
    ]
    ]
    The matrix will be 4x4, with columns as the transformed standard basis vectors e1, e2, e3, and e4.

  9. Maple

    1. refpkg


      Question: How do I get the routines in this package to run on my machine at home?
      Answer: Detailed instructions are here.

    2. lsqpkg

    3. misc



  10. Lectures on the Internet


    1. Audio Files


    Question: My system doesn't have a sound card. What should I do if I want to listen to the audio over the web?
    Answer: My advice: buy a soundcard. But if you can't afford one, you can get a driver that will allow you to play real audio through the built-in speaker on your machine. Try one of the many ftp archives on the web, such as Washington University Archive.

  11. Slides