Homework Statement
I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce.
Homework Equations
\vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) =
-\nabla\cdot (\rho\vec{u})\cdot\vec{e} -...
Homework Statement
I am having problems understanding the differential form of the conservation of mass.
Say we have a small box with sides \Delta x_1, \Delta x_2, \Delta x_3.
The conservation of mass says that the rate of accumulated mass in a control volume equals the rate of mass going in...
Homework Statement
Let ||\cdot || denote any norm on \mathbb{C}^m. The corresponding dual norm ||\cdot ||' is defined by the formula ||x||^=sup_{||y||=1}|y^*x|.
Prove that ||\cdot ||' is a norm.
Homework Equations
I think the Hölder inequality is relevant: |x^*y|\leq ||x||_p ||y||_q...
Here's my try. I'm using wolframalpha for the differentiation and integration...
\frac{dh}{dT}=\frac{d}{dT}\left(\frac{f(T)}{g(T)}\right)
=\frac{d}{dT}\left(\frac{1.28T}{378-3.16T}\right)
=\frac{4.04T}{(378-3.16T)^2}+\frac{1.28}{378-3.16T}
So now I have the change in time...
Hi,
a question at work popped up and it's been too long since I went to school :p
The total energy [Wh] required to heat the system to temperature T is given by f(T)=1.28T. The effect [W] applied to the system is given by g(T)=378-3.16T. How long does it take to heat the material to say 80...
Homework Statement
Prove that if m<n, and if y_1,\cdots,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j]=0 for j=1,\cdots,m. What does this result say about the solutions of linear equations?
Homework Equations...
Ah, I certainly missed the point of the question!
Let v be in the subspace spanned by y and z.
Then v=ay+bz for some numbers a and b.
But x+y+z=0 so z=-x-y.
v=(a-b)y+(-bx), that is, v is a linear combination of y and x and so is in the span of y and x.
This proves that the subpsace spanned by...
Homework Statement
Here's a statement, and I am supposed to show that it holds.
If x,y, and z are vectors such that x+y+z=0, then x and y span the same subspace as y and z.
Homework Equations
N/A
The Attempt at a Solution
If x+y+z=0 it means that the set {x,y,z} of vectors...
Since m+n \in L,\;n\in L and L is a subspace (closed under vector addition), we know that m \in L?
From (L \cap N) I know that n \in L, and from (L \cap M) I know that m \in L.
m+n must also be in L since it is a subspace.
Now, m+n \in (M+(L \cap N)) and m+n \in L and so x is an element of...
From our assumption that x\in L \cap (M+(L\cap N)) , we have that x\in L.
Since x=m+n we have that m\in L and n \in L , so
L \cap (M+(L\cap N)) \subset (L\cap M)+(L\cap N) .
Is that it or do I have to show that (L\cap M)+(L\cap N)\subset L\cap (M+(L\cap N)) ?
Suppose x \in (L\cap...
Homework Statement
Suppose L, M, and N are subspaces of a vector space.
(a)
Show that the equation
L \cap (M+N) = (L \cap M)+(L \cap N)
is not necessarily true.
(b)
Prove that
L \cap (M+(L \cap N))=(L \cap M) + (L \cap N)
Homework Equations
N/A
The Attempt at a Solution...
"Partitioned Orthogonal Matrix"
Hi,
I was reading the following theorem in the Matrix Computations book by Golub and Van Loan:
If V_1 \in R^{n\times r} has orthonormal columns, then there exists V_2 \in R^{n\times (n-r)} such that,
V = [V_1V_2] is orthogonal.
Note that...
Ah, I should have stated that, thank you.
You are usually the one that answers all questions I post around here. It's incredible that you do that for free..You should set up a paypal account :)
Thanks again.
Homework Statement
(a) If n is even find a polynomial function of degree n with n roots.
(b) If n is odd find one with only one root.
Homework Equations
N/A
The Attempt at a Solution
If by no roots, they mean no real roots then I guess:
f(x) = x^n+1 would work for both even...
Hi,
Here's my try at the n=3 case:
\begin{tabular}{ r c l }
\(f(x)\) & \(=\) & c_3x^3+c_2x^2+c_1x+c_0\) \\
& \(=\) & c_3x^3-c_3x^2a+c_3x^2a+c_2x^2+c_1x+c_0\) \\
& \(=\) & (x-a)c_3x^2+c_3x^2a+c_2x^2+c_1x+c_0\)
\end{tabular}
c_3x^2a+c_2x^2+c_1x+c_0 is a polynomial of...
Since we show that f(x) can be written as (x-a)g(x)+b for the n=1 case, we can assume that it holds for k \leq n, just as you wrote in an earlier post. Then we check to see if the statement holds for the n+1 case. I understand that (to the extent I can understand anything).
f(x)=c_1x+c_0
if...
Hi,
could someone explain how we go from 1. to 2. in the expressions below?
I fail to see how c_{n+1}x^na=(x-a)r(x)+k_1
Thanks.
\begin{align*}
1.f(x) &= (x-a)(c_{n+1}x^n) + c_{n+1}x^na + (x-a)q(x) + k_0\\
2.f(x) &= (x-a)(c_{n+1}x^n) + (x-a)r(x) + k_1 + (x-a)q(x) + k_0\\
\end{align*}
Thank you :blushing:
One more question (kind of the same):
The article goes on with the induction step:
Now we assume that this is true whenever d<k and let d=k, so that
m=n+k. Let f_1=f-(\frac{a_m}{b_n}x^{m-n}g).
I do not understand this last step.
Since \frac{a_m}{b_n}x^{m-n}...
(Thread should be named: Question about proof of the division algorithm, sorry about that)
Hi,
I am reading this proof of the division article:
http://xmlearning.maths.ed.ac.uk/lecture_notes/polynomials/division_algorithm/division_algorithm.php" [Broken]
I will write some of it here in case...
Ok, so my new function is,
f(x) = \sum^n_{i=1}a_if_i(x)
if I put x_i into this new function, I would get:
f(x_i) = a_1+a_2+...+a_n
The problem asks for a function where f(x_1)=a_i . Does this imply a sum over the a_i's?
By the way, I am in no way saying your answer is wrong...
First of all, I know that this thread is very old, but since I am working on this exact problem I assume it is better not to create a new thread. (+ it shows that I did a search :) )
Here's my attempt:
f_i(x) = \prod^n_{\frac{j=1}{j\neq i}} \frac{x-x_j}{x_i-x_j}
The next part of this...
Here's my complete solution. I expand and simplify the equations given in my first post.
Then I put together expressions for the x's and separate the constants.
I'm sorry about the formatting.
Does this look ok?
\[ f_{1} = k_{12}(x_{2}-x_{1}-l_{12}) = k_{12}x_{2} - k_{12}x_{1} -...
Homework Statement
Suppose masses m_{1}, m_{2}, m_{3}, m_{4} are located at positions x_{1}, x_{2}, x_{3}, x_{4} in a line and connected by springs with constants k_{12}, k_{23}, k_{34} whose natural lengths of extension are l_{12}, l_{23}, l_{34}.
Let f_{1}, f_{2}, f_{3}, f_{4} denote the...
We know that the row space is in R^{n} and that it is orthogonal to the null space.
Imagine that we have a 2x3 matrix with rank 2. It's row space would be a plane in R^{3}, and it's null space a line perpendicular to that plane. If we pick a point x in that plane, wouldn't the point closest to...
CompuChip: Does mathematica have a drawing tool? Say you want to explain how projection matrices work, and would like to draw a plane and some vectores. Is this easily done in Mathematica?
Asymptote looks interesting and quite hard to learn..
Thanks for the reply.
Hi,
What do you guys use for creating graphs and other figures for use in math/physics papers?
Is there some industry standard being used by science book writers?
Thanks.
I notice that many universities use Java as an introduction to programming and CS.
Stanford has a free online Java class, but I would not recommend it as they use custom libraries. This makes it harder to get help at various discussion forums.
MIT uses Python and they have an entire class...
I think it helps to think of span just as Mark44 explains it.
In R^3 a vector such as <1, 0, 0> would span a line, in this case the x-axis. You can find any point on the x-axis by multiplying the vector with a scalar; 3<1, 0, 0> gives you a point (3,0,0) on the x-axis.
Then if you have two...
Maybe it would help if you see it like this: (I'm no good with latex, sorry).
\left[ \begin{array}{ccc} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & -1 \\ a_{31} & a_{32} & 1\end{array} \right] [ \begin{array}{ccc} a_{0} \\ a_{1} \\ a_{2}\end{array} ] = 0
It follows from the fact that det I = 1.
By eliminating entries above the pivots in your upper triangular matrix, it can be made into a diagonal matrix.
Since we also know that the determinant is a linear function of each row/column separately, we may factor out:
a_{i,j} for i = j you get...
Hi,
I am currently half-way through Strangs Introduction to Linear Algebra, Second Edition.
Together with his online lectures, assignments, exams and other goodies, I feel it's a nice way to learn the subject.
After looking through a couple of other books, I get the idea that Strangs...
Homework Statement
With conductances c_{1}=1, c_{2}=c_{3}=2, multiply matrices to find
A^TCAx = f .
For f = (1,0,-1) find a solution to A^TCAx = f .
Write the potentials x and currents y = -CAx on the triangle graph, when the current source f goes into node 1 and out from node 3...
Homework Statement
Use Schwarz inequality on \bar{v} \bullet \bar{w} to prove:
||\bar{v} + \bar{w}||^2 \leq (||\bar{v}|| + ||\bar{w}||)^2
Homework Equations
Schwarz inequality:
|\bar{v} \bullet \bar{w}| \leq ||\bar{v}|| ||\bar{w}||
The Attempt at a Solution
The way I...
Hi Billy Bob, thanks for the reply.
Here's the way I think you would do it: (just showing one matrix)
\left[
\begin{array}{c}
A & A\\
\end{array}
\right]
\left[
\begin{array}{c}
R & 0\\
\end{array}
\right]...
Homework Statement
Prove that the three matrices have the same rank.
\left[
\begin{array}{c}
A\\
\end{array}
\right]
\left[
\begin{array}{c}
A & A\\
\end{array}
\right]
\left[
\begin{array}{cc}
A & A\\
A & A\\
\end{array}
\right]
Homework...