Spring 2012 Math 235 Homework from Bretscher's Linear Algebra(3/4e) which 
we'll go over in the 2:30 class - !ed problems are challenges for you to 
think about - these assignments are always subject to change as necessary.
Unless notified otherwise, the Bretscher problems will NOT be collected.
[Problem numbers are from 3e - numbers for 4e indicated where possible.]

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Starting week 2, there are also WeBWorK problems for you to do online:

	http://webwork.math.umass.edu/webwork2/MATH_235_4_7_S12/

regardless of your section number.

[Recall that your Username is the stuff before "@student.umass.edu" and
your (initial - change it!) Password is your UMass student ID number.]

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Supplemental Instruction sessions for Math 235 take place in 

	W. E. B. DuBois Library 1349
	???day 7:15-8:30PM & ???day 7:15-8:30PM

and there should also be tutors available there from time to time.
(Please check directly with the SI folks for details.)

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SPECIAL INFO:  This is being reconstructed - please excuse the potholes!
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			*** MINI-MIDTERM EXAM1 ***
 	         *** Monday 13 February 2012 in class ***

			*** JOINT MIDTERM EXAM ***
 	   ********** Wednesday 7 March 2012 at 7pm **********
 		**********     Hasbrouck 20    ***********
   	[Make-up: please contact profkusner@gmail.com in advance - 
		  10-noon Thursday 8 March 2012 in LGRT ????] 
	
			*** MINI-MIDTERM EXAM2 ***
 	           ***        ?????????????           ***

       			   *** FINAL EXAM ***
 	  ***** ***day ?? May 2012 at ??? in ????????????????? *****
	[as with MIDTERM: one (2-sided, 8.5x11") sheet of notes only]
    [Make-up (only if pre-approved by profkusner@gmail.com well in advance: 
          10-noon ???day ?? May 2012 in LGRT 1322/1334?]

			      ***REVIEW***	
	Rob ???
     ???????

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OLD PRACTICE FINAL: http://www.math.umass.edu/~norman/235_f10/exam/m235pracfinal.pdf

PRACTICE MIDTERM:  http://www.gang.umass.edu/~kusner/class/235f11pracmid.pdf

ANSWERS:  http://www.math.umass.edu/~norman/235_f11/exam/pracmidans1.pdf
[and then replace 1 with 2 or 3 to see answers to others]

USEFUL LINK TO OLD EXAMS: 
http://www.math.umass.edu/~markman/math235_fall08_html/practice-exams.html

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MORE SPECIAL INFO:  This schedule is still being revised!!!!!!
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	[There are copies of Bretscher ON RESERVE in the MAIN LIBRARY]

        [Please read indicated sections even if no problems are assigned]

01/25	[The following is a *true* story, from the summer of 2010:]

        I walked into the post office in my father's village one summer
        and was greeted by 3 similarly dressed young ladies of whom I first 
        asked "Are you all sisters?" and then realized one was mom and the 
        other two were her daughters (hey, I'm getting old ;-).  So while
        mom was picking up a package, I tried entertaining the kids with 
        a little magic trick of "guessing" their ages.  First I "guessed" 
        mom's age (29, of course ;-), but then I needed a bit of information
        to "guess" the others.  So I asked the bigger daughter if she could
        ADD her age to her sister's and tell me the answer: the SUM was 10. 
        Then came the hard part: how much older are you than your little 
        sister?  She thought for a bit and answered: the DIFFERENCE was 4.
        Then I explained that meant twice the bigger sister's age was 14, so 
        she must be 7 (Yes! the older one squealed), and her little sister 
        must then be 3 (Yes, the little one shyly smiled).  And then the
        mom (who's attention I'd been cleverly trying to get ;-) exclaimed: 

                             That's Linear Algebra!
        
        [And I did **NOT** reply "OK, let's work this out in detail with 
        pen and paper" - but *you* guys can and should do that now if it 
        wasn't already clear from class (no need to turn this one in) ;-]
	
	Note: the same ideas - in fact, the very same equations! - could
	be applied to a more serious situation, say, where there was a 
	mixture of two unknown ingredients (like electrons and positrons)
	but only the sum (total mass) and difference (total charge) could
	be measured, yet we needed to know how much there was of each 
	ingredient (to prevent the universe from collapsing in minutes ;-).
  
                                *       *       *

        An (infinite) chessboard can be thought of as the integer points
        in the plane: in other words, a possible chess move is of the
        form mH + nV where H and V are horizontal and vertical moves
        by a single square, and where m and n are (positive or negative)
        integers.  

        Analyze the various moves of different chess pieces.  For example,
        a rook (castle) can make a basic H or V move, and thus can move to 
        any position on the board - how about a knight?  How few knight 
        moves does it take to make a basic H or V move?  

        Bishops move diagonally, and thus (unlike the other pieces) 
        they always stay on their same color.  These moves are of the 
        form mH + nV where m+n is even; alternatively, they are of 
        the form aA + bB where A = H + V and B = -H + V are basic 
        diagonal moves.  Explain why these descriptions of bishops'
        moves are equivalent (see next problem for one way to go
        about this - this last part is what I want you to turn in).
 
                                *       *       *

        The atoms in salt (NaCl) are located at the points of the integer
        (cubic) lattice in 3-space:

                lI + mJ + nK  where l, m and n are integers, and
        
        where I, J and K are translation vectors along the axes.

        If we place a Na atom at the origin O, we have seen that
        there are two descriptions for the sites of the Na atoms:

        (*)     lI + mJ + nK  where l+m+n is an EVEN integer,

        (**)    aA + bB + cC  where a, b, and c are integers, and
        
        where A = I + J, B = J + K, C = I + K are diagonal vectors
        on the faces of the cube.  

        We want to see that these two descriptions of the Na sites
        (also known as the face-centered cubic, or FCC, lattice) are
        equivalent.  To do this, find a formula 

                F(a, b, c) = (l, m, n)

        which gives the location in description (*) of a point V = 
        aA+ bB + cC in description (**).  Also find an inverse formula
        
                G(l, m, n) = (a, b, c) 

        which lets us recover the coefficients in (**) from those in (*).

        As a warm-up, first work out the values of a, b and c you will need
        to make I + 2J + 3K, in other words, solve the vector system (***) 
        of 3 equations in the 3 unknowns a, b and c so that 

        (***)             aA + bB + cC = I + 2J + 3K

                                *       *       *

        This was a problem that the NPR "Car Guys" posed last week as a 
        Puzzler (their answer may be posted on line, but it may be too
        late to hear that over the airwaves - please do it on your own):

        Cats cost $1, dogs cost $15, and mice cost only $0.25.  I want a 
        herd of 100 animals, and I have $100, so being a cat lover, it's
        tempting to get only cats and mice.  But I also want a dog, and
        my cat(s) want mice to "play" with, so the numbers of dogs, cats 
        and mice are each positive.  And no cash left over.  What's the 
        solution? 

                                *       *       *	
	
	Also, please read the Preface/TextOutline in Bretscher.

02/01	1.1 # 10, 12, 16, 18, 24
     	1.2 # 10, 18, 22, 24, 36

What happens if we append the row

        10      11      12

to the matrix

        1       2       3

        4       5       6

        7       8       9

from class?  Its reduced row echelon form has the same 2 first rows 
as the original:

        1       0       -1

        0       1       2

(and the remaining two rows are all 0s).  Therefore, the corresponding
system of 4 (homogeneous) equations in the three unknowns r, s and t
still has the same set of solutions as the original (and not only the
trivial solution r=s=t=0).

What would have happened if we had appended the row

        10      11       c

with c anything other than 12?  And what happens if we keep appending
more and more rows like

	13	14	15

and so forth?

                                *       *       *

        Here's a fun problem that involves not only linear algebra, but 
        also elementary ideas about divisibility or number theory:

        Suppose you want to make $5 using exactly 100 common US coins.  
        Easy, you say: just use nickels (5 cents each).  But I say, no - 
        we have NO nickels, only pennies (1 cent), dimes (10 cents) and 
        quarters (25 cents) - is it possible to make $5 now with exactly 
        100 of these coins?  Why or why not?

[HINT: Row reduce the (augmented) matrix

	1	1	1   |	100

	1	10	25  |	500

and recall that we want (positive) integer solutions, corresponding to
the number P of pennies, D of dimes, and Q of quarters.]

02/08   Challenge problem: How many rrefs of an m by n matrix are there? 
	[Hints: try the square (m=n) case first, then distinguish the cases 
	m>n and m<n.]
	1.3 # 4, 6, 8, 10, 18, 24, 34, 36, 52, 56 

02/15	2.1 # 6, 10, 12, 20, 26, 42
        2.2 # 2, 10, 16, 32, 36, 40   

	*** MINI-MIDTERM this week in class on Chapters 1 & 2(part) ***


02/22 	2.3 # 2, 6, 20, 36, 48
	2.4 # 6, 14, 20, 26, 32, 44, 48-49[4e:80-81], 86[4e:104]

	[Hints for problems in 2.4#.... 

	32: (This problem has "disappeared" from 4e!)

	In #31 (as in the example I did in class) we saw there can be 
	infinitely many solutions to BA=I_2.  Are there any solutions
	to AB=I_3 for this example?  How about for *any* example? The
	hint in the book is called "reductio ad absurdum" or "arguing
	to a contradiction": if it were possible for AB=I_3, then any
	vector x would map under the composition AB to AB(x)=I_3(x)=x; 
	on the other hand, following the hint in the book, find (why?) 
	a nonzero vector x with B(x)=0, thus AB(x)=0 (why?); this 
	contradiction implies our assumption (what?) was impossible.
	
	48:

	Since P0 + P1 + P2 + P3 = 0, the center (of mass) of the tetrahedron 	     is at the origin 0.

	Note that the basic vectors e1, e2 and e3 occur at the midpoints of
	the edges, as do -e1, -e2 and -e3.  In fact ei = (P0 + Pi)/2 for i =
	1, 2, 3.

	So, for example, if M (this is one I made up) is the symmetry which
	interchanges P0 and P1, leaving P2 and P3 fixed (i.e. a reflection
	in the plane containing 0, P2 and P3), we see

  	  M(ei)	= (M(P0) + M(Pi))/2  [why?]
	
	  (i=1)	= (P1 + P0)/2 = e1
	
	  (i=2) = (P1 + P2)/2 = -e3

	  (i=3)	= (P1 + P3)/2 = -e2  

	and thus the matrix for M is

		|1	0	0 |
		|		  |
		|0	0	-1|
		|		  |
		|0	-1	0 |

	You can use this idea to get matrices for all the symmetries (there
	are 24 altogether!), including those sought in the problem as stated.
               
	86:

	a) The coefficients in the formulas for (I,L,S) in terms of (R,G,B) 
	give the matrix for P;  b) the matrix which cuts out the blue light 
	is a 3-by-3 projection matrix; c) & d) try making diagrams for the
	various linear maps and their compositions, like we did in class.


02/29	Find symmetry matrices for various letters of the Roman alphabet.
	[Please observe that the letter "O" can have infinite symmetry
	when written most symmetrically, while "R" has only the identity I,
	and "B" has two symmetries (I and a reflection R).  The letter "X"
        has eight eight - I, C, C^2, C^3, D, DC, DC^2 and DC^3 -
        you should write C and D down explicitly and explain what
        each does to the plane; e.g. which are reflections...?
        To actually *see* what the symmetries do, you may wish to
        color the corners of the "x", or label then with "1","2","3" and "4"!
	All the remaining letters are less symmetric....]


03/07	3.1 # 2, 4, 6, 10, 16, 20, 24, 40, 48, 53!, 54!
	
	[Hints for #48:
	a) write w=Au and explore Aw=A(Au)=...
	b) show that only possibilities for A in each case are...
	c) v=(v-Av)+Av decomposes v into a ker(A)-part and an im(A)-part] 

	3.2 # 2, 6, 10, 20, 24, 34   

	A question adapted from a "CAR GUYS" puzzler:

	Suppose you want to spell out your room number

		TWELVE

	with reflective metallic letters that adhere to your door.

	You go to the hardware store and find that three of your housemates
	want to do the same thing with their doors.

	Now the different letters have different prices, and you want to
	predict what yours will cost, based upon the total costs for each of
	theirs

		ONE	= $2

		TWO	= $3

		ELEVEN	= $5

	So... What does your TWELVE cost?!?!?!?

	[Hint: One systematic way to do this is to think of each word
	as a vector in R^26.  With this in mind, try to express the
	vector TWELVE as linear combination of ONE, TWO and ELEVEN.]

	[Comment: There is not enough information to pin down the price
	of each letter, but as long as you come up with a consistent guess
	for such prices, you will arrive at the same cost for your TWELVE
	- explain!  Try to generalize this puzzler in some interesting
	way: for instance, what is the smallest collection of spelled-out
	NUMBERS (with their total costs) in order to compute the prices
	of each letter in (a subset of) the alphabet?]

03/14	3.3 # 2, 18, 28, 30, 38, 48[4e:68], 56[4e:76]!, 62[4e:82]!
	3.4 # 2, 8, 20, 32!, 36!, 40, 48, 50!, 56, 60 	

	[Hints on HW in 3.3 # 

	30	This subspace is also ker([2  -1  2  4]).


	48	Get practice by trying dimensions you can visualize:
		in R^2 two 1-dimensional spaces can intersect in either
		a 0- or 1-dimensional space (ppoint or line); in R^3 two
		two dimensional spaces can intersect in either a 1- or 2-
		dimensional space (line or plane); in R^3 a 1- and a
		2-dimensional space can intersect in either a 0- or
		1-dimensionalspace.... 

		What about a 2- and a 3-dimensional space in R^4?

		Let's call these V and W.  Problem 47 suggests how to
		answer this.  First, the sum of their dimensions 

			dim(V) + dim(W) = 5

		must equal the sum of the dimension d of their intersection
		plus the dimension of V+W = span{V \union W}.  Now V+W is
		subspace of R^4, so its dimension is at most 4.  We are after
		the number d, the dimension of their intersection, and the
		formula from 47 becomes

			5 =< d + 4

		or d >= 1.  On the other hand, the intersection of V and
		W is a subspace of both V and W, so its dimension is at most
		the min of the dim(V) and dim(W), i.e. d = <2.  Thus

			d = 1 or 2 are the only possibilities.

		These are realized, by the examples

			V = span{e1, e2}, W = span{e2,e3,e4}

		and 

			V = span{e1, e2}, W = span{e1,e2,e3}

		respectively.  

		Now try the same kind of reasoning for the problem at hand.

	
	56	Again, try a case where n is small to get intuition:
		for n=2, the example

			      |	0	1 |
			A =   |		  |
			      |	0	0 |

		works, with v1 = e1, Av1 = e2.

		If you now just follow the hint in the book, and apply 
		A^[M-1} to the relation sugested, you'll get the equation

			A^{m-1}(c0 v) = c0 A^{m-1}(v) = 0

		which can only happen if c0 = 0, since we assumed
		A^{m-1}(v) is NOT zero (this defines "nilpotent
		of order m" - the first example has order m = 2).

		What power of A would you apply to the relation to
		get c1 = 0?
	
	
				- Good luck!]

[There's a BRAND NEW practice midterm and answer sketches at my webpage!]

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[I have checked the 3e versus 4e problem numbering through here....]
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

MIDTERM INFO:

See top of this page or my ~/classes.html page for practice midterm!

Here's a link to an old practice midterm exam:

	http://www.gang.umass.edu/~kusner/class/m235_fall08pracexam.pdf

Here is a link to an even older practice midterm exam:

	http://www.math.umass.edu/~kusner/class/old235pracexam.pdf

And at the end of this file are answers to T/F questions from each 
chapter end, which may help you to review concepts for the midterm.  
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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	For the rest of the semester we plan to cover the following topics:
	Vector spaces, linear maps and coordinates  (Chapter 4)
	Orthogonality:  Gram-Schmidt (5.1) and inner product spaces (5.5)
	Determinants (6.1, 6.2) 
	Eigenvalues, eigenvectors (chapter 7)

	For those of you not familiar with differential equations,
	here's where you can learn a little more:

 	    www.math.umass.edu/~kusner/class/diffeqnotes.pdf

	("Quick and Dirty Differential Equations" notes by Peter Norman,
	with just a little help from yours truly ;-)

03/28	4.1 #4, 8, 14!, 24, 26, 36, 40!, 50, 52, 58!  [... but do it early!]

04/04	4.2 #2, 4, 6, 17!, 20, 22, 23!, 64, 65!, 78!
	4.3 #2, 4, 5!, 15!, 18, 23!, 29!, 48, 49!

	Read Chapter 5.  Think about the relation between orthogonality
 	and linear independence.  How do you find "coordinates" of a vector
        with respect to an orthonormal basis?  We'll discuss the Gram-
        Schmidt process in detail very soon....    

04/11	5.1 # 6, 10, 12, 18, 28, 36!,40!, 42!
 	5.2 # 4, 6, 20!, 32, 34!, 40!     
	
        In class we will have discussed an inner product <p,q> on the 
        polynomials in x: <p,q> is the integral of p(x)q(x) dx from -1
        to 1.  A basis for the quadratic polynomials is {1, x, x^2}.
        Use the Gram-Schmidt process to convert this into an orthonormal
        basis with respect to this inner product.

	[Section 5.5 touches on this also.]	

	5.3 # 4, 10, 16!, 18!, 20!, 28, 31!, 34!, 37! 
	5.5 # 2, 4, 8, 9!

	[Instead, we will cover some of this material in a special session
	for anyone who is interested at 10AM Sunday 5 December in LGRT 101] 

	Read through chapters 6 & 7 over the Thanksgiving break.

04/18	6.1 # 2, 8, 14, 24, 26, 32, 38!, 46!, 52!	
	6.2 # 2, 8, 12, 14, 16, 18!, 20!, 26!, 30

04/25	7.2 # 6, 12, 16, 20, 28
        7.3 # 4, 16, 26
	7.4 # 2, 12, 14, 40!, 48!, 58!
	7.5 #

REVIEW:

	We'll meet the ????day before the final exam to review (see above). 

FINAL EXAM:

	*****????day ?? May 2012 at ??? in ????? ???????*****


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(Old) Practice Exam:  http://www.gang.umass.edu/~kusner/class/pracexam2.pdf

	Also, please think about this problem:

	Suppose we want to compute the exponential of a matrix

                exp(A) = I + A + A^2/2 + ... + A^k/k! + ...

        - how does having a basis of eigenvectors for A, along with
        their corresponing eigenvalues, help us?!        

	Practice final solutions (not sure I like the sound of that):

	    http://www.math.umass.edu/~kusner/prac235finalsols.pdf  

HAVE A GREAT SUMMER BREAK!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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Answers to T/F questions from the end of each chapter for Bretscher
_Linear Algebra, 3E [of course, it is YOUR job to figure out why].
!!!!!!!!!!!!!!!!!! Warning: 4E has changed !!!!!!!!!!!!!!!!!!!!!!!!!
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============================no guarantees===========================
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Chapter 1

1       T
2       F
3       F
4       T
5       T

6       F
7       F
8       F
9       T
10      T

11      F
12      F
13      T
14      T
15      T

16      T
17      T
18      T
19      F
20      F

21      F
22      T
23      F
24      T
25      F

26      T
27      F
28      F
29      F
30      T

31      T
32      T
33      F
34      T
35      F

36      T
37      T
38      T
39      F
40      F

41      T
42      T
43      T
44      F
45      T


Chapter 2

1       T
2       F
3       T
4       T
5       F

6       T
7       F
8       F
9       T

10      T
11      F
12      T
13      T
14      T
15      T

16      T
17      F
18      T
19      F
20      T

21      F
22      T
23      F
24      T
25      T
26      T
27      T
28      F
29      F
30      T

31      F
32      F
33      T
34      T
35      T

36      T
37      F
38      T
39      F
40      T

41      T
42      F
43      T
44      F
45      T

46      T
47      F
48      F
49      F
50      F

51      F
52      T
53      T
54      T [over the COMPLEX NUMBERS; what about REAL NUMBERS?]
55      T

56      T

Chapter 3

1       F
2       T
3       T
4       F
5       T

6       F
7       T
8       F
9       T
10      F     

11      T
12      T
13      F
14      T
15      T

16      F
17      T
18      T
19      T
20      T

21      F
22      F
23      T
24      T
25      T

26      T
27      F
28      F
29      T 
30      T

31      T
32      F
33      F
34      T
35      F

36      T
37      T
38      F
39      F [although there is a natural way to realize it as a subspace!] 
40      T

41      T
42      F
43      T
44      T
45      T

46      T
47      T
48      F
49      T
50      T

51      F
52      T
53      F [this one is tricky ;]

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============================no guarantees===========================
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