Difference between revisions of "HWK1MA527Fall2013" - Rhea

Revision as of 14:29, 27 August 2013

Homework 1 collaboration area

Feel free to toss around ideas here. Feel free to form teams to toss around ideas. Feel free to create your own workspace for your own team. --Steve Bell

By the way, I sign my name at the end of a post by typing three tildes in a row, and the Rhea puts my name in for me. - Steve Bell

Here is my favorite formula:

$ f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz. $

This is a test formula:

$ A \vec x= \vec b $ - Eun Young

People have noted an error in the solutions in the back of the book. This from Jeff R.:

For Section 7.2 problem 29, they have the profit vector as

[85, 62, 30]

but the problem defines it as

[35, 62, 30]

which obviously gives a different answer. Just wanted to make you aware of the mistake. (Thanks, Jeff. Steve Bell )

• Be Careful!

Question from a student :

I have a question on the Kirchoff law problem, section 7.3 problem 18. On the loop portion of the defining equations, relative to a clockwise direction, I find the equation of the right loop to be -12 I2 + 8 I3 = -24. Is this correct to assume the I2 term is negative due to the counterclockwise flow of I2, as with the voltage term?

Answer from Eun Young :

Yes, it's correct and it's same as $ 12 I_2 -8 I_3 = 24. $

Remark from Steve Bell :

I remember from my engineering days that engineers make one convention about the sign of the increase in voltage around a loop in Kirchoff's Laws and physicists make the opposite convention, so this might be a cultural thing. The important thing to understand here is the math.

Question from a student :

That equation is the same thing I got, I just took a different direction for my KVL around the loop (e.g. $ 12 I_2 - 8 I_3 = 24 $). However, I don't fully understand what your question is. I do however have a question on p. 287 #12, 14, and 15. Is the book looking for a rigorous proof or just an example of this property? --Ryan Russon 18:48, 25 August 2013 (UTC)

Answer from Eun Young :

When a problem asks you to show or prove something, you need to provide proof. When a problem asks you to disprove something, you need to give an example. Hence, you need to prove #12, 14, and 15. There is a theorem about rank in sec 7.4. With this theorem, you can show the properties in #12, 14, and 15 easily.

Remark from Steve Bell :

Perhaps the word "proof" is overkill here. I would like you to be able to explain in words why something is a general fact, much like I do in class. It doesn't need to feel like a mathematical proof, but it does need to be convincing.

From --Ryan Russon

Thanks for the clarification. I was on my way to working on a very rigorous proof and then wondered if all the work was necessary.

Question from a student :

I have a question on the last 2 homework problems in section 7.4 #32, 34. How exactly do we determine if something is a vector space? Do all vectors have to be linearly independent? I know in class we went over a couple of test to determine if vectors are a subspace. Does that come into play? Thank you. From --Katie Mathews

Answer from Eun Young :

In #32, we basically want to see if $ \{ (v_1, v_2, v_3 ) \in R^3 | 3v_1 - 2 v_2 + v_3=0 \text{ and } 4 v_1 + 5v_2=0 \} $ is a vector space. Note that this set is a subset of a vector space $ R^3 $. So, you just need to show whether it is a subspace or not. #34 is similar. In general, to determine if a set with operations is a vector space, you need to check 8 axioms. You can find those on [1].

Remark from Steve Bell:

Yes, you'll find those vector space axioms in section 7.9. The good news for us now is that we don't need to check all those axioms because our sets are subsets of $ {\mathbb R}^n $. We only need to check two things.

1. Is the set closed under addition?

2. Is the set closed under multiplication?

That's the Subspace Test.

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