Linear Definition and 1000 Threads

I want to understand which of these is computationally expensive (in the sense of computational time) which is more accurate. Also I want to understand which of these two problems (computations time + accuracy) of iterative methods are addressed by multi-grid methods?

Hey! :o We have the vectors $\overrightarrow{a_1}=\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}, \overrightarrow{a_2}=\begin{pmatrix}-1 \\0 \\ 2\end{pmatrix}, \overrightarrow{a_3}=\begin{pmatrix}7 \\ 8 \\ 6\end{pmatrix}$. I have shown that these vectors are linearly dependent: $\begin{bmatrix}...

Hello PF, I have just finished my first semester in college and did Calc. 3. Now for the spring semester i have to take differential equations and i have been given the advice that linear algebra comes in handy when dealing with DEs. So can anyone recommend a good introduction for linear algebra...

Hi! First off, I am actually a math / econ major. I hope I'm still welcome here I am trying to figure out if it's worth it to take both of these courses or just one of them. I have not taken LA before. Course 1: Addition, subtraction and scalar multiplication of vectors, length of vector...

We were going over linear independents in class and my professor said that if y1 and y2 are linearly independent then the ratio of y2/y1 is not a constant, but he never explained why it is not a constant.

Hi I've collected few sets of data and obtained significant different linear regression (R^2) in 2 particular sets of data . Does that indicates the 2 sets of data is not validated which might due to data collection error? For example, 20 sets of data contain linear regression of 0.900+...

In a problem I am working on, it is given that $V_1, V_2, ... , V_n$ are mutually perpendicular vectors in a space defined with a certain scalar product. I need to prove or disprove that $V_i$ are linearly independence regardless of any definition of scalar product. I think the solution should...

Homework Statement A very long cylinder of linear dialectric material is placed in an otherwise uniform electric field ##E_0##. Find the resulting field within the cylinder. (The radius is ##a##, the susceptibility ##X_e##, and the axis is perpendicular to ##E_0##) Homework Equations Boundary...

Homework Statement Consider a two-layered cylindrical wire with inner-layer permeability μ1 and outer-layer permeability μ2. A line current I runs through the center in the z direction. Calculate the bound currents and the magnetic field produced by the bound currents. Homework Equations [1]...

In the image above, a centroid with radius 1 is depicted. F1 is pointing directly at point A (which is the center of the circle), and F2 is pointing directly at point B. The radius for finding the torque would be the perpendicular between the center of the object and the force vector, so r1...

Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...

xy'' + 2xy' - y = 0 Honestly no clue where to start, Wolfram Alpha gives a rather complex answer lol (http://www.wolframalpha.com/input/?i=xy%27%27%2B2xy%27-y%3D0)

Hello all. I have a question about linear dependency. Suppose we have a set ##S## of functions defined on ##\mathbb{R}##. ##S = \{e^x, x^2\}##. It seems very intuitive that this set is linear independent. But, we did something in class I'm unsure about. Proof: Let ##\alpha, \beta \in...

Homework Statement Homework Equations The Attempt at a Solution I set up the Wronskian?? and compare them?

Good evening! I'm having trouble completing an assignment related to inelastic collisions and momentum. Homework Statement Here is the statement: You are in great shape on this hot July morning. You are working as an intern for the Montreal Police (SPVM) Collision Investigation unit. You...

if A and B are 3x3 matrices such that: ABC = I, |3A|=81 and |C^T|= 2 , how to find |B^-1| I couldn't solve this because there is not much given.

Homework Statement Does a linear transformation ##g : \mathbb{R}^2 \rightarrow \mathbb{R}^2## so that ##g((2, -3)) = (5, -4)## and ##g((-\frac{1}{2}, \frac{3}{4})) = (0, 2)## exist? Homework EquationsThe Attempt at a Solution For a linear transformation to exist we need to know if those two...

The problem I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ## $$ \begin{cases} x = 1 + s - t \\ y = 2 - s \\ z = -1 + 2s \end{cases} $$ The attempt ## s, t ## are the parameters for the two directional vectors which "support" the plane. $$...

Hi! I have a question about Linear Energy Transfer for electrons. According to the definition the linear energy transfer is equal to the energy dE which a charged particle loses at a distance dl. But what does dl mean in case of electrons? Is LET calculated per unit of full path length or per...

We learned that the condition of a linear transformation is 1. T(v+w) = T(v)+T(w) 2. T(kv) = kT(v) I am wondering if there is any transformation which only fulfil either one and fails another condition. As obviously, 1 implies 2 for rational number k. Could anyone give an example of each...

$\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...

sin(x) * y''(x) + 2cos(x) * y(x) = 0 y(0) = 0 y'(0) = 1 how do I solve it? (I know the solution because I have created the diff. equation starting from a simple function). -- lightarrow

Homework Statement A particle of mass m moves along the x–axis under the influence of force ##F_x=-ax+bx^3## , where a and b are known positive constants. (a) Find, and sketch, the particle's potential energy, taking U(0) = 0 (b) Identify and classify all equilibrium points (c) Find the...

Homework Statement Show that any vector in a vector space V can be written as a linear combination of a basis set for that same space V. Homework Equations http://linear.ups.edu/html/section-VS.html We are suppose to use the 10 rules in the above link, plus the fact that if you have a lineraly...

m1v1 = m2v2 v = rω m1(rω)1 = m2(rω)2 m1ω1 = m2ω2 Does this make sense?

Homework Statement A 15 000kg loader traveling east at 20km/h turns south and travels at 25km/h. Calculate the change in the loader's a) kinetic energy b) linear momentum Homework Equations ΔKE= 1/2mvf^2 –1/2mvi^2 Δp= mv2 –mv1 m= 15000 vi = 20km/h = 5.56m/s vf= 25km/h = 6.94m/s The Attempt at...

Suppose that I have an overdetermined equation system in matrix form: Ax = b Where x and b are column vectors, and A has the same number of rows as b, and x has less rows than both. The least-squares method could be used here to obtain the best possible approximative solution. Let's call this...

Hello, I'm trying to develop a pendulum to test protective equipment so I want to work out the length I'll need to generate a desired velocity and the necessary mass I'll need for a specific moment of inertia. I know there are multiple ways to solve for linear velocity with equating Ek and Ep...

1. Homework Statement Posted Homework Equations Posted The Attempt at a Solution Posted I just need the work to be checked.

What are the best ones and why for a first-timer like myself (doing self-study)? Thanks very much everyone.

I am working on a problem that goes like this: Show that $Ker (F) \cap I am (F) = \{0\}$ if $F: W \rightarrow W$ is linear and if $F^4 = F.$ I have the solution but there is one step which I need help: (the delineation is mine) (1) Suppose that there exists $x$, such that $x \in Ker(F) \cap...

hi there,im pretty new to antenna design, still have a lot of theory to go through, so please be kind. ;)I have to design a linearly polarized microstrip patch antenna (rectangular) that is fed with 2 coaxial probe feeds. The patch should be a square (L=W), so i can place 2 feeds orthogonally...

Homework Statement (This is a mock-up) Given maximize(x,y) Ax + | x - y | s.t. x + By - C >= 0 y >= 0 A, B, and C are some coefficients. Homework Equations Linearizing absolute expressions: z = | x |; where z >= x, z >= -x Linearizing free variables: let x = x+ - x-; where x+, x- >= 0...

Hi PF! The other day in class my professor mentioned something about expanding linear independent vectors, but he did not elaborate. From what I understand, if ##x_1,...,x_k## are linearly independent vectors in ##V##, where ##dimV=n>k##, how would you extend ##x_1,...x_k## to a basis ##\{...

Homework Statement Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...

Homework Statement 3y(t)+2=x(t) Homework Equations k1y1(t) + k2y2(t) + 2(k1+k2) = k1x1(t)+k2x2(t) The Attempt at a Solution I know the system is non linear but I cannot explain why. It has something to do with 2(k1+k2) but I am unsure.

I'm looking at purchasing Algebra (2nd Edition) by Michael Artin, is this a good book to purchase as my first intro to linear algebra book for self learning?

1.) Two volumes of an encyclopedia stand side by side, in order, on a shelf. A bookworm starts at page i of Volume I and bores its way in a straight line to the last page of Volume II. Each cover is 1 mm thick, and the first volume is 2/5 as thick as the second volume. The first volume is 40 mm...

Homework Statement Find the general solution for the DE: t2y''-2y=0 Homework Equations These were given for other parts of the problem so I'm not sure if they're relevant. y1(t)=t2, y2(t)=t-1, y(1)=-2, y'(1)=-7 The Attempt at a Solution The t2 at the front was really stumping me and I'm not...

I was given a question relating centripetal force to solve A linear spring of negligible mass requires a force of 18.0 N to cause its length to increase by 1.0 cm. A sphere of mass 75.0 g is attached to one end of the spring. The distance between the centre of the sphere M and the other end P of...

So anyone of you know a book that provides a gentle and quick refresher for linear algera, in the spirit of the book "Quick Calculus" by Kleppner and Ramsey? Now that I am studying quantum mechanics, I feel I need to review the linear algebra I studied during my engineering degree. Thanks.

Hi all, I have a question regarding a simple crank and slider mechanism, but I'm not a mathematician. This means that long formulas scare me... (: I've been searching around and haven't found anything I can really understand. I'm trying to get a "ball park" formula to calculate the amount of...

Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...

Homework Statement A semi-circle of radius R has a charge Q uniformly distributed over its length, which provides a line charge density λ. Determine E at the origin. Homework EquationsThe Attempt at a Solution https://www.physicsforums.com/attachments/105239 I can tell by argument of...

<1 and <2 form a linear pair. If m<1 = 5x + 9 and m<2 = 3x + 11, find the measures of both angles.

May I start by saying hello to all and that I have visited the site before, but have yet to create an account until now. With that said, I am having difficulty calculating proper numbers for a project I am designing. Mind you, it has been about seven years since I was in any physics or...

Homework Statement Construct a subset of the x-y plane R2 that is (a) closed under vector addition and subtraction, but not scalar multiplication. Hint: Starting with u and v, add and subtract for (a). Try cu and cv Homework Equations vector addition, subtraction and multiplication The...

I have the following linear differential equations: ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0## Where x and y are functions of t, and A through D are constants. I can solve this fairly easily by differentiating the first equation, rearranging, and removing one of the variables, which gives me a...

Hi, I have some questions about the 2D SWE (Shallow Water Equations). I would like to know if an exact (analytical) solution is available in the case of the (linear) surface "gravity" wave. Indeed, my example of interest is given by a square [0,L]2 with initial velocity field u = v = 0 and an...

So many moons ago I made myself a pretty bog standard dual linear bench power supply using an LM317 and LM337 to give me -15 - 0 - +15. In an attempt to get more current out of it (for no real reason other than because I can) I got a larger transformer and a beefier rectifier, then using the...