Work area for Practice Final Exam questions
Question:
For problem 1 on the practice problems, is the reason the answer is D is because det(A) has infinitely many solutions?
Answer:
No, that's not the reason. 3x3 matrices form a vector space, so the question as to whether or not the set of 3x3 matrices A such that det(A)=0 is a vector space is asking if this set is a subspace. There are only two things to check for the subspace condtion:
1. If v_1 and v_2 are in the set, is their sum in there too?
2. If v is in the set, is cv also in for any constant c?
Matrices with zero determinant satisfy 2, but not 1. It is pretty easy to come up with two matrices that fail. Let's see ...
1 0 0 0 1 0 0 0 0
and
0 0 0 0 0 0 0 0 1
for example.
Follow up question: Then why is iii) a vector space. I can think of some symmetric 3x3 matrices that have determinant = 0. For example:
1 0 0 0 0 0 0 0 1
I don't understand how the set of ALL symmetric 3x3 matrices could be considered vector spaces?
Question:
Can anyone fill in the blanks on the last problem (23) Professor Bell worked in class today?
I follow up to the u(x,t) = 1/2(sin2x)(cos2t) + ......
Where did the 1/2(sin2x)(cos2t) come from?
Answer:
When we did the method of separation of variables to solve the string problem, we got solutions of the form
X_n(x) T_n(t).
Then we took linear combinations and realized what the coefficients had to be from plugging in the initial conditions.
The cos 2t term is the T(t) part of the solution that goes with sin 2x. (The B_n coefficients are zero because there is zero initial velocity.)
The part of this problem that makes the final answer so short and sweet is that cos pi/2 =0, cos 3 pi/2 =0, etc. , cos (odd) pi/2 =0 makes all those terms in the infinite sum go away.
Follow up Question: I understand the separation of variables, the X_n(x) T_n(t) solutions, and the linear combination. I don't understand how the f(x)is used to get the u(x,t)=1/2 sin2x cos2t......This isn't making any sense. Also, how do we know what L is in this problem?
Question:
Can someone explain the purpose of the infinite sum 1/n^2 in problem 30? I understand how to use the Parseval's identity, but that last term in the problem statement is really confusing me.
Answer:
When you square the coefficients of the Fourier series, you get four times the sum of 1/n^2.
Follow up question: Then were did the 4 go? Wouldn't the 4 then get divided into the RHS making the answer D instead of C?
Question:
Has anyone had any success with Problem 15? I keep solving this on and getting the solution A. I know I'm not doing it correctly. Any hints?
Answer: Your e^-2t term should be e^-2(t-1).
(Recall L(u(t-a)f(t-a))=e^-as*F(s)
