Problem Sets for E3101: Applied Math I: 2008

Just do the assigned problems.
Unless noted, all problems are from Strang: Introduction to Linear Algebra: 3rd Edition.
Notation: problem 1.1.4, e.g. refers to section 1.1, problem 4
"Review Problems" are given in parentheses and are only suggested problems for the TA to do in office hours/recitations.
Matlab Problems are all optional. (but recommended as a good way to check your homeworks and learn something very useful)

 
All Homeworks are Due by 5pm one week after they are assigned (unless otherwise noted):
 Drop off in homework box in DAPAM (200 Mudd)

Late Policy --50% credit if handed in 1-day late, 0% after that.
Talk to TA's directly for specific exceptions

HOMEWORK REQUIREMENTS

  • DO YOUR OWN WORK!
  • Please PRINT your name on the homework
  • Please staple your homeworks and write clearly (and darkly) so the TA's can see your work.
  • Please show your work. No credit for the solutions without supporting calculations will be given.
    Thank you for your consideration
     
    Week Problem set Problems
    1 1
    due 11 Sept
    		 1.1.1, 1.1.3, 1.1.5,1.2.1, 1.2.3, 1.2.5 (see/review 1.2.20), 1.2.13 (hint: a picture might help) ,2.1.10,2.1.11 (review 2.2.13, 2.2.14)
    (Optional Matlab exercises: check the red problems using matlab (but still do them)...if you're having trouble distinguishing the colors, let me know)
    Here is a pdf file of the Problem Set from the Book (many thanks to Matt Davis)
    2
    		 2
    due 16 Sept
    		 2.2.8, 2.2.12,  2.2.13, 2.2.26, 2.3.1,2.3.3 (also, if b=(1,0,0)  what is the vector  c such that Ux=c? hint: use what you already know) , 2.3.17, 2.3.28 (practice mini proof), 2.4.5,  2.4.14, (and explain your answers)(Review  2.2.14,,2.3.10 (good exercise),2.3.25, 2.4.15, 2.4.16), Extra Credit: 2.2.27: solve for s by setting up the appropriate 4x4 system and using elimination.

    (Matlab exercises in red plus 2.2.21,2.2.29.  For 2.3.17 check the calculation and plot the solution and the points!)
    3
    		 3
    due 25 Sept    		 use Gauss-Jordan elimination to prove the inverse formula for a general 2x2 matrix A=[ a b ; c d],2.5.9, 2.5.27, 2.5.10 (hint: think block or permutation matrices) ,2.5.29, 2.5.30,2.6.7, 2.6.13 (also factor as LDLT), 2.6.16, 2.7.16 (include AB),2.7.22 (review 2.5.7, 2.5.12, 2.5.23,2.6.10,2.6.17,2.6.19,2.7.7,2.7.19,2.7.24,2.7.39 (good theory question) (Matlab exercises to check are in red plus 2.6.32, 2.6.33).
    4
    		 4
    due Tues
    Oct 7
    		3.1.10,3.1.17,3.1.18,3.1.19,3.1.20,3.1.23,3.1.27,3.2.1 (and continue to the reduced row rechelon form R),3.2.2,3.3.2
    (review 3.1.9,3.1.22,3.1.24,3.1.28,    3.2.12,3.2.21,3.2.23,3.2.27,find special solutions for 3.3.2)
    5
    		 5
    due Tue
    Oct 21
    		 3.4.1,3.4.5,3.4.10,3.4.25,  3.5.2, 3.5.10 (hint: turn the equation into Ax=b and think), 3.5.17, 3.6.3,4.1.11,4.1.14 (good and tricky), 4.2.1,4.2.3, 4.2.11 (review 3.4.4, 3.4.7,3.4.13 this is a good one to think about, 3.4,31,3.4.34,3.5.1,3.5.5,3.5.11,3.5.14 (interesting),3.5.27,3.6.2, 3.6.17,3.6.23,3.6.28,4.1.21)
    6
    		 6
    due Tue
    Oct 28    		 4.2.22, 4.2.29, 4.2.30 (extra credit: prove this result for all rank one matrices), Least Squares Problems (Know how to do these) 4.3.1,4.3.9,4.3.27,4.4.5,4.4.6,4.4.7,4.4.10,4.4.15, 4.4.18 (Review  4.1.16, 4.2.5, 4.2.13 which is quick and useful, 4.2.21, 4.2.17, 4.3.10,4.3.17,4.3.18,4.3.22,  4.3.26, ).
    7
    		 7
    due Wed
    Nov 5
    		 Some goodies on Gram-Schmidt and the QR 4.4.20,4.4.21 (also: find QR such that A=QR and show that the projection is given by p=QQTb), 4.4.24, 4.4.33 (Fun with the determinant) 5.1.1,5.1.3,5.1.24, 5.1.27,5.1.28,5.2.17 (good matlab check here),5.3.17 (Review   4.4.23, 4.4.34,4.4.35,5.1.18,5.2.18,5.3.8)
    8
    		 8
    due Thur
    Nov 20    		 6.1.2,6.1.6, 6.1.9 , 6.1.25 (this is a nice problem...see if you can find a general solution for the eigenvalues (and eigenvectors) of all rank 1 matrices..i.e. matrices that can be written as uvT where u and v are any two column vectors, hint: see problem 6.1.13), 6.1.30 (good practice here),6.2.1, 6.2.5 (easy but important) 6.2.10,6.2.20,  6.2.24 (review,5.1.19,6.1.8,6.1.10,6.1.13,6.1.14,  6.1.28, 6.1.35 )
    9
    		 9
    due Mon
    1 Dec
    		 6.2.34, 6.2.39 (googlish problem),6.3.1, 6.3.3,6.3.4, A R&J problem: do opposites attract? 6.3.11,6.3.13, Fun with symmetric matrices  6.4.5, 6.4.19, 6.4.21,6.4.24
    (Review    : 6.2.2,6.2.22,6.2.23, 6.2.35, (good theory question), 6.3.19,6.3.24 (good practice in clear thinking but assume A is diagonalizable), 6.4.3, Matlab: learn to use the ODE integrators ode45 to solve general dynamical systems numerically)
    10
    		 10
    due Tues Dec 9    		 Last Problem Set  
    • Prove that if A has rank r, then ATA is also rank r (and therefore there are exactly r non-zero singular values)
    • Fun with symmetric matrices and the SVD. 6.4.25, 6.5.6,6.5.28, 6.7.4, (also look at 6.7.5 or use this result to get the SVD),6.7.7,  7.4.2,7.4.3,7.4.6  (these last three are really a set) the pertinent section on the pseudo-inverse starts on page 395.) 7.4.19 (hint: AA+ and A+A are projection matrices: this is a good problem to understand the relationship between the SVD and the 4 sub-spaces). 
    • use the SVD to show that x+=A+b is a solution of the normal equations (i.e. that ATAx+=ATb) even if A is singular. (therefore x+ is a least-squares solutions...and the shortest one at that)
     (review 6.4.13, 6.4.24,    6.5.25,6.7.7,7.4.17,7.4.20) (Matlab: 6.4.28 (this is sort of fun), 6.4.29)

    Back to syllabus

    marc spiegelman
    Last modified: 13 Oct 2008