Fall 2008 Math 235 Homework from Bretscher's _Linear Algebra_ (3e) due each 
THURSDAY in class - !ed problems need not be turned in but are for you to 
think about - assignments are always subject to change as necessary:

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Final Exam schedule:

Thursday 18 December at 8AM in Agricultural Engineering Building 119

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

Review: We plan to meet Sat 13 Dec 1:30PM in E-lab 306 (usual classroom)

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NOTE :  Joint mid-term exam on Thursday 16 October at 7PM in Marcus 131 
	  - the exam will cover the material through section 3.2.

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


	[There are 4 copies of this text ON RESERVE in the MAIN LIBRARY]

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

09/04	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).
 
			*	*	*

	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 (*).

	Also, please read the Preface/TextOutline in Bretscher.

09/11	1.1 # 10, 12, 16, 18, 24
     	1.2 # 10, 18, 22, 24, 36

09/18   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 

09/25	2.1 # 6, 10, 12, 20, 26, 42
        2.2 # 2, 10, 16, 32, 36, 40   

10/02 	2.3 # 2, 6, 20, 36, 48
	2.4 # 6, 14, 20, 26, 32, 44, 48-49, 86

[NOTE: these problems can be turned in to me Tuesday 10/07]

	[Hints for problems in 2.4# : 

	32:

	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.


10/09	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....]


	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] 


      [NOTE: No class Tuesday 10/14 - work on the practice exam problems:
	http://www.gang.umass.edu/~kusner/class/m235_fall08pracexam.pdf] 


10/16	3.2 # 2, 6, 10, 20, 24, 34   [MIDTERM this evening - see below!]

	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?]


MIDTERM:  Joint midterm exam this Thursday 16 October at 7PM in Marcus 131 
	  - the exam will cover material through (part of) Chapter 3.

	[make-up: Friday???]

Here's a link to the practice midterm exam:

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

Here is a link to an old practice midterm exam:

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

And here are answers to T/F questions from the end of each chapter,
which may help you to review concepts for the midterm:

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

Bretscher _Linear Algebra (3e) "True or False" Exercise Solutions
[Of course, it is YOUR job to figure out why!]

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 ;]

====================================================================
============================no guarantees===========================
====================================================================
 
After the MIDTERM: please read the rest of Chapter 3 - also...: 

10/23	3.3 # 2, 18, 28, 30, 38, 48, 56!, 62!
	
	[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!]


10/30	3.4 # 2, 8, 20, 32!, 36!, 40, 48, 50!, 56, 60 

	For the rest of the semester we plan to cover the following topics:

	Linear spaces (chapter 4)
	Orthogonality:  Gram-Schmidt (5.1) and inner product spaces (5.5)
	Determinants (6.1, 6.2) 
	Eigenvalues, eigenvectors (chapter 7)

[Rob will be away 11/06 in Indiana - a "guest lecture" will be 
given by Dr. S. Mehrotra (mehrotra@math.umass.edu) - I'll collect
all the chapter 4 homework on 11/13...]

11/06	4.1 #4, 8, 14!, 24, 26, 36, 40!, 50, 52, 58!  [... but do it early!]
	
11/13	[We *do* meet on Wednesday 11/12 as well!]

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

	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 ;-)

		 	
	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....    


11/20	5.1 # 6, 10, 12, 18, 28, 36!,40!, 42!

        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.]	

 	5.2 # 4, 6, 20!, 32, 34!, 40!     
	5.3 # 4, 10, 16!, 18!, 20!, 28, 31!, 34!, 37! 
	5.5 # 2, 4, 8, 9!

	[Maybe these later sections will be due Tuesday 11/25]

11/27	THANKSGIVING: gobblegobblegobblegobblegobble(d up a week of HW)
	[Read chapters 6 and 7 over the holidays (between bites of turkey)]

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

12/11	7.2 # 6, 12, 16, 20, 28
        7.3 # 4, 16, 26
	7.4 # 2, 12, 14, 40!, 48!, 58!

	********************REVIEW SESSION************************

We'll plan to meet sometime SaSuMoTu before the final exam.

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  


FINAL EXAM: ???