Homework Statement
Show that x^{n}+y^{n}=z^{n} has a nontrivial solution if and only if the equation \frac{1}{x^{n}}+\frac{1}{y^{n}}=\frac{1}{z^{n}} has a nontrivial solution.
Homework Equations
By nontrivial solutions, it is implied that they are integer solutions.
The Attempt at a Solution...
Homework Statement
Let V be a finite dimensional vector space and T is an operator on V. Assume μ_{T}(x) is an irreducible polynomial. Prove that every non-zero vector in V is a maximal vector.
Homework Equations
μ_{T}(x) is the minimal polynomial on V with respect to T.
The Attempt at...
I think I get it now. The function maps the coset of V mod U to the same coset of V mod U by mapping each individual element to another element in that coset.
I'm sorry if it seemed I brushed over your post and didn't completely read it. I did. I just didn't understand it. I'm having trouble...
This is where I confuse myself. T is a linear transformation from V to V. If it maps the set v+U to the set v+U, then wouldn't that be mapping cosets of V mod U to cosets of V mod U instead of mapping V to V?
Homework Statement
This question came out of a section on Correspondence and Isomorphism Theorems
Let V be a vector space and U \neq V, \left\{ \vec{0} \right\} be a subspace of V. Assume T \in L(V,V) satisfies the following:
a) T(\vec{u} ) = \vec{u} for all \vec{u} \in U
b) T(\vec{v} + U) =...
What I take away from the First Isomorphism Theorem is that if two vectors \vec{v}, \vec{u} \in V, then for any Linear transformation T:V\rightarrow W,
if T(\vec{u}) = T(\vec{v}) then \vec{u} \equiv \vec{v} modulus Ker(T)
So if you only take the cosets of V mod Ker(T), then it follows that...
We went over all three of those. I don't fully understand them, though. I think that's my main problem. I went back to basis because I was familiar with that.
It is "the collection of cosets of V modulo Ker(f)".
I quoted that from the book because I would have gotten that wrong. I understand...
Homework Statement
Let V be a vector space over the field F and consider F to be a vector space over F in dimension one. Let f \in L(V,F), f \neq \vec{0}_{V\rightarrow F}. Prove that V/Ker(f) is isomorphic to F as a vector space.
Homework Equations
L(V,F) is the set of all linear maps...
The sum of a geometric series is defined as:
a+ar+ar^2+ar^3+...+ar^{n-1} = a\frac{1-r^n}{1-r}
If n started at 0, then a would be 2.
Since n starts at 1, in order to form a geometric series we must group it as following:
\frac{8}{3} + \frac{8}{3}(\frac{4}{3}) + \frac{8}{3}(\frac{4}{3})^2 +...
I'm assuming that second 2 is a typo and should be an n.
\sum ^{14}_{n=1} 2(\frac{4}{3})^n
I believe the equation is working. a represents the first term in the series. In this case, what is a?
If Kernel(T)=Kernel(T^2), then Range(T)=Range(T^2)
First I started by saying that
Kernel(T) = Kernel(T^2) \Rightarrow Nullity(T) = Nullity(T^2)
By the Rank Nullity Theorem we have the following:
Dim(V) = Rank(T) + Nullity(T)
Dim(V) = Rank(T^2) + Nullity(T^2)
\Rightarrow Rank(T) +...
Thank makes sense!
Then it follows that since
\text{ker}(T) \subset \text{ker}(T^2) and
Nullity(T)=Nullity(T^2)
then \text{ker}(T) = \text{ker}(T^2)
Since the Kernel is, by definition, a subspace of V. And if a Vector Space with dimension n is contained in another Vector Space with...
Ah, ok. I think that makes sense. Thanks!
Ooh. I didn't understand that operator on V implied T:V->V.
Does being an operator imply that it is a linear transformation? Or is that part just assumed in the problem?
Ok. So trying that I get
T^2(c_{1}\vec{v_{1}}+c_{2}\vec{v_{2}}) =...
I hope I'm posting this in the right place.
Homework Statement
Let V be a finite dimensional vector space over a field F and T an operator on V. Prove that Range(T^{2}) = Range(T) if and only if Ker(T^{2}) = Ker(T)
Homework Equations
Rank and Nullity theorem:
dim(V) = rank(T) +...
Edit: Nevermind. Now I'm curious though.
I might just be going out on a limb here, but I think that looking at it from this perspective might help:
By the definition of logarithm,
log_{10}n = x
implies
n = 10^x
I hope it helps at least
I think the best way to approach this problem is to visual what the problem is asking for. What is the region under the Normal Curve that it wants?
In this case, the symmetry could be of use to you.
This still won't work. ceil(rand*5) will only produce integers 1-5.
In your 2nd while loop, shouldn't that only be counting the number of times each die has rolled? It seems like your 2nd while loop encompasses much more than you want it to.
In your program you set the counters d1-d6 to zero...
Velocity is a vector. Since the velocity is downwards, it is negative. When you took the square root to solve for the final velocity, you wrote + instead of +/-
That's right, and I believe your math is correct too.
Since all three sides of the triangles are congruent:
\overline{DB}\cong\overline{DC} given
\overline{BA}\cong\overline{AC} because they are the same length
\overline{AD}\cong\overline{AD} trivial, they share a common line
Then the...
The acceleration of gravity on Earth is always -9.81m/s^2. An object speeding at 3000m/s towards the ground will feel the same -9.81m/s^2 acceleration as an object that has just been dropped. In reality, an object going at 3000m/s might be slowing down if the air resistance is large enough, but...
If the stones are thrown vertically downward, you can think of the angle being 90° (the angle between the positive x-axis and the negative y-axis). The horizontal component of the velocity (magnitude*cos90) = 0 and the vertical component of the velocity (magnitude*sin90) is just the original...
The simplification in the equation could have also been obtained by grouping the four corners on the Karnaugh map instead of just boxes 0 and 8.
I didn't see anyone else mention it, so I thought I would throw that out there :smile:
This will not generate a fair die. The quantity rand*5+1 will produce a number in the range 1-6. It sounds like it would be fair, but rounding the number will produce the following results:
1 (1 <= x < 1.5) - 10% of the time
2 (1.5 <= x < 2.5) - 20% of the time
3 (2.5 <= x < 3.5) - 20%...
The Kinetic Energy is equal to the work that the friction force must do to stop the box. Since W=Fd, the distance that box travels is d=W/F. So yes, you did it correctly.