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date:	Mon, May 15, 2017 at 12:20 PM
subject:	Grades are posted on website, to be submitted soon

I have posted course grades to a course website, see

http://www-users.math.umn.edu/~adams005/MATH4604/distributioncourse.txt

I plan to submit the grade to the registrar at the end of the day
today (Monday), or, maybe, early tomorrow morning.

If you have questions or concerns, it would be good if we could
discuss them before grades are submitted. Feel free to contact me by
email, or to call me at 612-625-5507. I plan to be around my office
for most of the day today.

Thanks. You've been an excellent class, learning a lot of very
difficult material.

Best,
Scot
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date:	Mon, May 8, 2017 at 1:05 PM
subject:	Final exam

Hi,

This message is to confirm that the final exam will be on Thursday 11
May, this coming Thursday, 1:30pm-3:30pm, in the regular classroom,
Amundson 104.

Between now and then, I will hold my usual office hours.

Best,
Scot
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date:	Thu, Apr 13, 2017 at 11:28 AM
subject:	One more typo, in #48.

Bill pointed out another HW typo, this one in HW#48: Be sure that the
superdomain of g is equal to V, same as for f. I've fixed this in the
HW, but, as usual, you may need to clear cache and refresh your
browser to see the corrected version. Thanks, Bill.

Also, a hint for #48: To show that a=b, one approach is to prove
        both     a =^* b     and     b =^* a.
When I write up my solution to #48, I think I'll probably do the work by proving
    first, that             D_p((f,g))   =^*   (D_p f, D_p g)
    then, that     (D_p f, D_p g)   =^*   D_p((f,g)).

- Scot
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date:	Tue, Apr 11, 2017 at 3:38 PM
subject:	Change to #47

I made a couple of changes to #47, so please use the version online,
and not the one from class.

I forgot to bind the variable i.

Also, it turns out that
   \partial_i ( f ( p + bullet ) )
might not be equal to
   \partial_i ( f_p^T ),
because f(p) might not be defined, in which case f_p^T doesn't even
exist. So I just took out all reference to  f_p^T  from the problem.

- Scot
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date:	Mon, Apr 10, 2017 at 10:50 AM
subject:	Office hours canceled on Th 13 April

With regrets, I have to cancel office hours on Th 13 April.

Class is NOT canceled. We will have class at the usual time.

Best,
Scot
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date:	Mon, Mar 27, 2017 at 8:56 AM
subject:	Typo in HW#35

Sasha noticed a typo in HW#35. It shouldn't say
   ... = \emptyset.
It should say
   ... = \{ 0_{VW} \}.

I've corrected it, but you might need to clear cache and/or refresh
your browser to see the updated version online.

Sorry for the typo. Thanks for the correction.

- Scot
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date:	Fri, Mar 10, 2017 at 1:22 PM
subject:	Small change to HW#31

There was a small error in HW#31 that Bill noticed. I changed it, but
you may need to clear cache and refresh your browser to see the new
version.

Thanks, Bill.

Sorry for the mistake.

- Scot
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date:	Tue, Feb 7, 2017 at 3:53 PM
subject:	Homework hints

First, I need to make a correction.
--------------------BEGIN
Let m,n \in \N.
Let A be an n x m real matrix. Let u \in \R^m and v \in \R^n.
Then B_A(u,v) is, by definition,
       the dot product of  L_A(u)  with  v.
--------------------END
In class today, I started with this, then somehow corrected myself to write
       the dot product of L_A(v) with u.
That makes no sense, because L_A : \R^m ---> \R^n, and v \in \R^n.
Sorry.
========================================
In working the HW, you may use, without proof, the following fact:

--------------------BEGIN
Let m,n \in \N.
Let A be an n x m real matrix.
Let e_1,...,e_m be the standard basis of \R^m.
Let f_1,...,f_n be the standard basis of \R^n.
Then, for all integers i \in [1,m] and j \in [1,n],
     A_{ji} is the dot product of  L_A(e_i)  with  f_j.
--------------------END

Pf: Given integers i \in [1,m] and j \in [1,n].
We wish to show that A_{ji} = [L_A(e_i)] dot [f_j].

By definition of L_A,
for any v \in \R^m, the components of
      L_A(v)
are obtained by dotting the rows of A with v.

In particular,
the jth component of the vector L_A(e_i)
     is equal to
(the jth row of A)   dot   (e_i),

The dot product of L_A(e_i) with f_j
     is equal to
the jth component of the vector L_A(e_i),
     which is equal to
(jth row of A) dot (e_i)
     which is equal to
the  ith entry in the jth row of A
     which is
A_{ji}.
QED

========================================
Remember that, for any vector u \in \R^k, by u^V we mean the
k x 1 column vector with the same entries as u. (The "V" indicates
that you should write the entries of u in a vertical way.)

Here's another useful fact that you may use in the HW (again,
without writing out a proof), if you want to:

--------------------BEGIN
Let m,n \in \N.
Let T : \R^m ---> \R^n be linear.
Let e_1,...,e_m be the standard basis of \R^m.
Let i \in [1,m] be an integer.
Then
                 [T]     is an n x m real matrix
and
             (e_i)^V   is an m x 1 column vector
and their product
                 [T] ((e_i)^V)
   is an n x 1 column vector that is equal to
                 (T(e_i))^V.
--------------------END

Pf: For any matrix A \in \R^{n x m},
the ith column of A is A((e_i)^V).

Thus
   [T] ((e_i)^V)
is the ith column of [T].

By definition of [T], the entries in ith column of [T] are
the entries in the vector T(e_i). If you put these entries into
a column vector, you get
   (T(e_i))^V.

Thus
   (T(e_i))^V
is the ith column of [T].

Therefore, [T] ((e_i)^V) = (T(e_i))^V, as desired.
QED
========================================

- Scot
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date:	Thu, Feb 2, 2017 at 2:16 PM
subject:	HW typos, link to last semeter

I corrected a couple of typos on the HW, so, if you've already printed
out, I guess you'll want to reprint it -- sorry. Also, be sure to
clear cache and refresh your browser, just to make sure you're getting
the newest version.

Also, to link to last semester's website, use the link in the
*leftmost* column, at the bottom, that reads
  MATH 4603 last semester
The URL is
  http://www-users.math.umn.edu/~adams005/MATH4603/
I won't archive that website until the summer, so it will NOT appear
in the right column under "ARCHIVE".

- Scot
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date:	Wed, Jan 18, 2017 at 8:54 AM
subject:	Correction to HW

In HW#3, I forgot the hypothesis that f be defined near a. That's
corrected now, but you might need to clear cache and/or refresh your
browser to see the new version.

Thanks, Bill, for pointing this out.

- Scot
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date:	Fri, Jan 13, 2017 at 12:05 PM
subject:	Room change for MATH 4604

The classroom for MATH 4604 has been changed. It is no longer in Peik Hall.

It is now in Amundson 104,
          on Tuesdays and Thursdays, 9:05am to 11am.
The time of the class has not changed, only the location.

See you on Tuesday at 9:05am in Amundson 104.

Best,
Scot
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