Linear Definition and 1000 Threads

I am working on derivation of Lorentz force. (I know that Lorentz force is in some sense definition of fields, but still there is nontrivial dependence on velocity). I want to derive that the force is linear in components of velocity, so for example $$F_x=q(E+Av_x + Bv_y + Cv_z ),$$where ##A...

I tried assumed ##\theta \approx sin \theta \approx tan \theta##. By Snell's law(after approximation), $$n_1 \tan( i_1)= n_2 \tan( i_2)$$ If ##\tan (i_1)=\frac {h_o}{ u}## and ##\tan (i_2)=\frac {h_i}{ v}##,then $$m=\frac {h_i}{h_o}=\mod{\frac {v n_1}{un_2}}$$ Which is the expected...

Hi, Could someone help me with this question: 2x – 1 = 9 – 3x Thank you in advance!

Hello, In grade 11 of high school, I encountered this linear programming problem on my textbook: The "alternative solution" described in the textbook as follows: Let: - ##x## : amount of plant A - ##y## : amount of plant S - ##L## : garden area - ##L_x## : area of garden for one plant A -...

My questions are now... Do the steps of converting this space to transfer function include any laplace ? or just we do get [SI-A]-1 and then transfer function is = C* [SI-A]-1 * B As [1 0] * [s-1/det -0.5/det ; 0.5/det s-0.5/det] * [0; 1] = -0.5/s^2+s+0.5 I mean do we need any laplace after that...

Hi, in the context of linear two-ports networks my (italian) textbook says that if its internal structure consists of passive one-port components with no independent current/voltage generators then the following (implicit) linear equations based representation always exist: ##\left[A\right] V...

I have no clue how to interpret this problem and where to start to get my values to plot my graphs and get my x, y and etc values as there is too much going on. Can someone shed some light of how to do this problem? I am not sure if is this how I solve this: How do I find that the HD...

Hi, my professor gave me I code where he used to evaluate the answer of a linear system to a step increase in the input variables like this: MySystem = ss(A, B, C, D); ltiveiw('step', MySystem, 'r-', 300); My problem is that with this code I get the response only for a positive step. I'd...

Hey! :o The code words of a linear code $C$ have the length $n=5$. Writing the code words into a matrix to get the linear independent ones, we get the following: \begin{equation*}\begin{pmatrix}0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 &...

Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$ $$a_1\left[ \begin{array}{r} 1\\-2\\0 \end{array}\right], a_2\left[ \begin{array}{r} 0\\1\\2 \end{array}\right], a_3\left[ \begin{array}{r} 5\\-6\\8 \end{array}\right], b=\left[ \begin{array}{r}...

2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...

Homework Statement Find the linear transformation [/B] T: R3 --> R2 such that: 𝑇(1,0,−1) = (2,3) 𝑇(2,1,3) = (−1,0) Find: 𝑇(8,3,7) Does any help please?

Given w = T (v), where T is a linear transformation and w and v are vectors, why is it that we can write any coefficient of w, such as w1 as a linear combination of the coefficients of v? i.e. w1 = av1 + bv2 + cv3 Supposably this is a consequence of the definition of linear transformations, but...

Attached is my work for the voltages, and the problem itself.

nmh{2000} 17.1 Let $T: \Bbb{R}^2 \to \Bbb{R}^2$ be defined by $$T \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 2x+y\\x-4y \end{bmatrix}$$ Determine if $T$ is a linear transformation. So if...

Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3. What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...

If I had to guess what the complex matrix would look like, it would be: ##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix} a+bi & 0 \\ 0 & -b+ai\end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}## I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover...

Are the vectors $$\left[ \begin{array}{r} 2\\1\\-2 \end{array}\right] ,\quad \left[\begin{array}{r} 0\\2\\-2 \end{array}\right] ,\quad \left[\begin{array}{r} 2\\3\\-4 \end{array}\right] $$ linearly dependent or linearly independent...

I need help to know if I'm on the right track: Prove/Disprove the following: Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0. (V is a vector-space) I think I need to disprove by using v = 0, however I'm not sure.

Dear All, Thank you for reading my post. I'm stuck with picking the correct 'generator' element in the attached example (Simplex method). As you can see, the solution keeps picking the yellow elements as 'generators', but I don’t understand why we can't choose the purple ones. Can somebody...

Good afternoon people. So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if: T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to...

I want to put myself a linear actuator on the sliding gate to the cottage. I looked in the internet, there are a lot of offers and the prices are very different. Maybe there are people here who understand this and can advise me what type of actuator is better, manufacturer and where to buy? I...

Summary: I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space. I need to do a linear model identification using least squared method. My model to identify is a...

ok this is a clip from my overleaf homework reviewing just seeing if I am going in the right direction with this their was an example to follow but it also was a very different problem much mahalo

Here is the initial matrix M: M = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \end{bmatrix} I have used the shortcut method outlined in this youtube video: LU Decomposition Shortcut Method. Here are the row reductions that I went through in order to get my U matrix: 1. R_3 -...

From this table: Looking at the 7 row and taking the first entry, 1 This is 000001 Adding back the highest and lowest bit: 10000011 And then the equation would just be x^7 + x +1 ? Or taking the taking the third entry, 7 This is 000111 Adding back the highest and lowest bit: 10001111 And...

##259x+581y=7## ##581=259.2+63## ##259=63.4+7## ##63=7.9+0## therefore by reversing... ##7=0.63+1.7## ##7=1.259-4.63### ##x=1## ##y=-4## is this correct? Bingo

On Ned Wright's pages one can find this graph: plotting some supernova data against different expansion models. The main thing here that gives me a pause is the linear relationship for the closed universe with ##\Omega##=2 (red line). There doesn't seem to be any weird scaling involved. What is...

Hi PF! What is meant by the spectrum of a linear operator ##A##? I read somewhere that if ##0## belongs in the spectrum, then ##A## is not invertible. Can anyone finesse this for me? I read the wikipedia page, but this was tough for me to understand. Perhaps illustrating with a simple example...

While doing some calculations I came across some terms which are ##\frac{(v•\vec \nabla)v}{a(t)}## and ##\vec \nabla•[\rho(1+\delta)v]## where all quantities have spatial dependence other than "a" which has only time dependence , the first term here is canceled and the the second term is...

Simplex method

Above is my attempt at a solution, however, this did not yield the correct answer. Any help is greatly appreciated!

Without providing further context, is it possible to give meaning to the construction of "nested linear subspaces"? e.g. in $\mathbb{R}^n$ the tangent space at a point $T \mathbb{R}^n = \mathbb{R}^n_x = V_0 \subset V_1 \subset ... \subset V_k $ is the same as saying that the tangent space can...

Solving using Linear Momentum: M vb2/2 = M g 2L vb = 2√(g L) m v = m v/2 + M (2√(g L) ) v = 4 M √(g L) / m Note: I see from the answers - that this is correct. -------------- Next, I tried to solve it via Energy conservation point of view. M vb2/2 = M g 2L vb = 2√(g L) m v2/2 = m v2/8 + k...

Hello, I'm trying to find the general solution of this homog. system w/ constant coefficients. I can't even get past the first step, which is to find the eigenvalues. As far as I know, there are a few approaches: 1) solve det(A-λI) = 0 2) solve the trace determinant plane equation (which is...

I am working with Magnetic strips to get countinous force in both clockwise and anticlock wise direction. I am looking for better concept for hand free mode in my experiment. The pdf file of my experiment is attached .

Hi I would like to find out please what it would mean to transform a vector based on some property that it has and if you do that to more than one vector would both operations be isomorphic in some respect. Is there a set of vector transformations of this time that could be used to process non...

So I've been looking at covalent bonds and come across the approx you can do of the molecular orbital for ##H^+_2## by just summing two 1s orbitals, the method is called the linear combinations of atomic orbitals, and you get what is below which I believe is exact in this case since the 1s...

I posted previously about this topic a couple of years ago, and it really came across like a ton of bricks, but now that I have established some credibility, perhaps it will be read with some interest. In the course of my career, (in the years 1986-1990), on two occasions I discussed with...

Hi all I am working on beam forming of high frequency (>400 KHz) linear array for imaging sonar. I have gathered sensor data of 80 channel linear array sub merged halfway in 8 m deep water tank with a transmitting probe 6 m away from it at same depth. I recorded multiple pulses and perform...

Does the Friedmann vacuum equation have a linear solution rather than an exponential one? Using natural units one can write Friedmann's equation for the vacuum as $$ \begin{eqnarray*} \left(\frac{\dot a}{a}\right)^2 &=& \frac{8\pi G}{3}\rho_{vac}\\\tag{1} &=& L^2 \left(\frac{\rho_0}{L^4}\right)...

Outside of QM, do perfectly linear waves really exist in nature? I am referring to just those waves that also have dispersion - in which the wave components have differing phase velocities. ... or are we just using the mathematics of linear systems to approximate nonlinear systems? Thanks in...

Consider a linear order (on ##\mathbb{N}##). Now it will have a comparison relation associated with it (a function from ##\mathbb{N}^2## to ##\{0,1\}##). Let's denote it as ##\prec##. I want to define a certain notion, which I will just call "uniformity" (just for brevity). For each ##i \in...

Hey i got a problem here but still without correction so if you guys can help me , thanks in advance I'm stuck there We have L : P -> R^2 L is a linear transformation with : B = \left\{1-x^{2},2x,1+2x+3x^{2} \right\} \; and \; B' = \begin{Bmatrix} \begin{bmatrix} 1\\-1 \end{bmatrix}...

Homework Statement Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set. Homework EquationsThe Attempt at a Solution...

So, I have no education in this field. I've really been into mutant breeding and crisper plasmid based genetic engineering. I've read that heavy ions create a greater number of mutagens than electron magnetic radiation such as deep uvc. lately particle accelerators have been on my mind. I'm...

I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...

Let A be a n x n matrix with complex elements. Prove that the a(k) array, with k ∈ ℕ, where a(k) = rank(A^(k + 1)) - rank(A^k), is monotonically increasing. Thank you! :)

I have already taken two elementary linear algebra courses, and have taken the upper-division linear algebra course offered at my school. However, I feel that I did not learn as much from the latter as I should have. I can owe this to not applying myself as much as I should have, due to other...

Homework Statement If I have an affine camera with a projection relationship governed by: \begin{equation} \begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b \end{equation} where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix...