Linear Definition and 1000 Threads

Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation and $$T \begin{bmatrix} 1 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 1 \\2 \\1 \\ \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \\ \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \\ \end{bmatrix}, \quad T...

Homework Statement This question was on a recent AP Physics 1 exam as a multiple choice; "Three air track gliders, shown to the right all have the same mass M. Gliders 2 and 3 are initially at rest. Glider 1 is moving to the right with speed v. Glider 1 collides with glider 2 and sticks to it...

show that $e^{2x},\quad \sin 2x$ are linear independent on $\left\{-\infty,\infty\right\}$ new concept to me but $\sin 2x$ has an amplitude $e^{2x}$ doesn't

Santa Fe is approximately 33.88 degrees north of the equator. Given that the Earth's radius at the equator is 3960 mi and the Earth spins around its axis completely in 24 hours, what is the linear speed of Santa Fe around the Earth's axis in mi/hr? I don't know where to start! Thank you so much!

I not very good at using the LaTex editor, so I took a photo of my HW questions. For the first question, I'm not really sure how to get started, should I write out a specific case? Like what would \varphi (P) be when m=1? For the second question, I know that a linear functional have two...

Anyone know of an alternative to the 7mm Drylin square linear bearing? They seem on the expensive side, 30GBP for just the plastic insert/bearing surface. https://www.igus.co.uk/product/983 Ideally I'm looking for something smaller, say for a 3mm shaft/rail?

Homework Statement "A rod of length L lies along the x-axis with its left end at the origin. It has a non-uniform charge density λ=αx where α is a positive constant. a) What are the units of α? b) Calculate the electric potential at A. Homework Equations Linear charge density: λ = Q/L where Q...

Homework Statement Am amount of charge Q is uniformly spread over a semi-circle of radius R whose center is located a distance A from the origin. What point charge would have to be placed at the origin so that the E field at the center of the circle is 0? (The open end of the semi-circle is...

Homework Statement Consider the below vector x, which you can copy and paste directly into Matlab. The vector contains the final grades for each student in a large linear algebra course. x = [59 70 83 89 72 70 54 55 68 61 61 58 75 54 65 55 62 39 43 53 67 100 60 100 61 100 77 60 69 91 82 71 72...

Homework Statement A system has a ball and a uniform rod. The rod is rotating about point X on a frictionless table until it strikes the ball. The rod stops and the ball moves away. Variables: Rod's mass: m1 Ball's mass: m2 Rod's original angular velocity: ω Ball's final velocity: v Rod's...

Hi, For a machine I am designing, the linear actuator would ideally not produce much noise as it is intended to be in a relatively quiet environment. Just wondering do all actuators produce a large amount of noise pollution (it would need to produce a force of up to 125kg so it is not a small...

I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ... I need some help in order to fully understand Example 6.76 ... ... Example 6.76 reads as follows: My question or...

Hi, I'm reading a book about numerical models for PDE and it says that a method is said to be stable if this condition holds: $$|| \mathbf u^{n+1} || \le c_t || \mathbf u^{n} ||$$ where ##c_t## is a constant greater than zero, and ##u## is the numerical solution to the problem. (In particular...

Designing a machine. For it to work properly there would need to be a mechanism that connects the linear actuator to the stand that would allow for the actuator to move horizontally relatively freely with applied forces at the bottom, when there is simultaneously a downward applied force by the...

Homework Statement Homework Equations For this problem I got the angular momentum conservation equations, mv(l+h)=mv'(l+h)+Ml2ω and momentum conservation equation as mv(l+h)=mv'(l+h) m=colliding mass,v and v' velocity before and after collision. M=mass of the rod. 2l=length of the rod...

Does anyone know a model to identify Straight Compact Linear data? I've been toying with Pearson Correlation Coefficient and am very disappointed. https://en.wikipedia.org/wiki/Pearson_correlation_coefficient I originally thought that this would be exactly what I needed, but... After some...

Dear all, I am trying to solve a question, and I think that something is missing. It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b. If the vector ku-3v is a solution to the same system, then: a) k = 4 b) k = 3 c) k = 0 The correct solution...

h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?

Homework Statement Sorry for all the posts lately. This should be the last one for a while. Solve the following system of linear equations: ##\displaystyle 4x-y=10## ##\displaystyle x+y-3z=8## ##\displaystyle 3x-y+z=12## Homework EquationsThe Attempt at a Solution [/B] I started with...

Homework Statement My problem has two parts. 1) We have two point masses ##m,M##. and there is another mass ##m_1## between them.They are all aligned in a line. Mass ##M## is moving with speed ##u_1## toward ##m_1## and after collision and all other masses are not moving. we want to find...

I want to model a sensor with the static behavior: y(t)=a+by_{0}(t) using a first order lag: G(s)=\frac{K}{1+Ts} However, if i try to convert this order lag in time domain and set the derivative as 0, what i get as static response is: y(t)=Ky_{0}(t)=by_{0}(t) And the a constant has...

I can't prove this proposition. I have however managed to prove that the linear combinations of the independent normal rv's are also normal by looking at it's mgf $$E(e^{X_1+X_2+...+X_n})=E(e^{X_1})E(e^{X_2})...E(e^{X_n})$$ The mgf of a normal distribution is $$e^{\mu t}e^{\frac{t^2...

Hello, I'm trying to get my head around linear transformations, and there are a few things I'm not grasping too well. I'm trying to understand combinations of linear transformations, but I can't find a lot of clear information on them. As far as I can tell, any two linear transformations of the...

I was musing about black hole mergers and what an observer might see for a particularly simple (i.e. blatantly contrived) scenario. Suppose that there are two (non rotating, un-charged, bog standard) black holes of dissimilar masses heading for a head-on collision. They will accelerate towards...

Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?

Problem: Suppose $V$ is a complex vector space of dimension $n$, where $n > 0$, and suppose that $T$ is a linear map from $V$ to $V$. Suppose that if $\lambda$ is any eigenvalue of $T$, then $ker(T−\lambda I)^2 = ker(T−\lambda I)$. Prove that $T$ is diagonalizable. Here's what I think I need...

Hey! :o Let $q$ be a power of a prime and $n\in \mathbb{N}$. We symbolize with $Tr$ the map of the trace from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, i.e. $Tr:\mathbb{F}_{q^n}\rightarrow \mathbb{F}_q$, $\displaystyle{Tr(a)=\sum_{j=0}^{n-1}a^{q^j}}$. I want to calculate the dimension of the image...

Homework Statement Determine whether the following system of equations has a single point of intersection. If so, find the point of intersection. 4x+y-9z=0 x+2y+3z=0 2x-3y-5=0 Homework Equations n1⋅(n2×n3) The Attempt at a Solution I have to pick a variable, use a pair of equations to...

Wooo! I am almost done with my stats class. I am having a reoccurring issue with finding the test statistic using a scatter plot and linear correlation coefficient. Internet Users (Per 100) Award Winners (Per 10 Million) 78.2 5.5 79.5...

Homework Statement Problem statement in attached photo. This is an ungraded assigned problem which I am using to study for an exam, so I don't need the whole solution just help with a couple of points I am confused about. One: Part d) is really important to how I will answer part b). If we can...

Given a finite-dimensional normed linear space ##(L,\lVert \cdot \rVert)##, is there anything that suggests that at every point ##x_0 \in L##, there exists a direction ##\delta \in L## such that that ##\lVert x_0 + t\delta \rVert \geqslant \lVert x_0 \rVert## for all ##t \in \mathbb{R}##?

Homework Statement "Let ##X,Y## be independent r.v.'s (EDITED) normally distributed with ##\mu=0,\sigma^2=1##. Find the distribution of ##W=2X-Y##. Homework Equations "If ##X,Y## are independent, then if ##Z=X+Y##, ##f_{Z}=\int_{\mathbb{R}} f_X(x)f_Y(z-x)\, dx##. The Attempt at a Solution...

Homework Statement We have an uncharged, conducting wire with radius a. We surround it by a linear dielectric material, εr, which goes out to radius b. We place this in an external electric field, Eo. Homework Equations We have electric potential inside (a < s < b) Vinbetween=Acosφ +...

Hello I have tried to resolve the problem below Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.040000 cm thick to a hemispherical dome with a diameter of 45.000 meters. My procedure was: the volume of the sphere is V=4/3 pi r^3...

Hello! (Wave) Let $F$ be a field with infinite elements and $V$ a $F$-linear space of dimension $n$ and $W_1, \dots, W_m$ subspaces of $V$ of dimension $n_i<n, i=1, \dots, m$. We want to show that $V \setminus{(W_1 \cup \cdots \cup W_m)} \neq \varnothing$. Fix a basis $\{ v_1, \dots, v_n\}$...

Can someone please check my work to this? The reflecting telescope is deployed in low Earth orbit( 600km) with each orbit lasting about 95 min. use the linear velocity formula to solve the problem. I did 300 * 95 min = 28500. Can someone check my work please? if someone could check it today...

Hello, I've been working through some Digital Signal Processing stuff by myself online, and I saw a system that I wanted to write down as a Linear Algebra Equation. It's a simple delay feedback loop, looks like this: The (+) is an adder that adds 2 signals together, so the signal from x[n]...

Hi PF! I'm solving a differential eigen-value problem in weak form, so I have trial functions. If the Wronskian of trial functions is small but not zero, is linear independence an issue? I have analytic trial functions but am numerically integrating.

Generalized momentum is covariant while velocity is contravariant in coordinate transformation on configuration space, thus they are defined in the tangent bundle and cotangent bundle respectively. Question: Is that means the momentum is a linear functional of velocity? If so, the way to...

Homework Statement If ##x## is a rational function of ##y##, such that (ax2 + bx + c)y + (a'x2 bx' + c') = 0 prove that (ac' - a'c)2 = (ab' - a'b)(bc' -b'c) Homework Equations The quadratic formula The Attempt at a Solution This equation can be rewritten as: (ay + a')x2 (by + b')x (cy +c') = 0...

I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc. But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc. I am trying to understand some of the subtleties of moving to 3D. In particular, for any...

The original problem for anyone that can read Chinese: https://zerojudge.tw/ShowProblem?problemid=b221 The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...

Homework Statement Not for homework, but just for understanding. So we know that if a matrix (M) is orthogonal, then its transpose is its inverse. Using that knowledge for a diagonalised matrix (with eigenvalues), its column vectors are all mutually orthogonal and thus you would assume that...

I have a question, let’s say I’m holding a long piece of wood such as. 1’ x 6’ plank and I’m rotating it in a circle by spinning around with my hands extended, I suddenly let go, what happens to the velocity of the wood since every point on the wood that is a different distance from the center...

If talking about a particle rotating around an axis away from it by r. if the particle is moving with constant angular velocity ω. is the linear velocity constant or no? Now what I know is that since we have Vt= ωr, so r doesn't change, as well as ω, so Vt is said to be constant. but I think...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focuses on Chapter 3: Elements of Point Set Topology ... I need help regarding a remark of Apostol's made after Definition 3.2 and Theorem 3.3 ...Definition 3.2 and Theorem 3.3 read as follows: In a note at the...

Homework Statement I can't understand an equation to LDA. The context is: Maybe the lecturer is trying to create a proof of the equation given below. I know the above that LDA projects the points along an axis so that we can have maximum separation between two classes.in addition to reducing...

We are studying 2nd and 3rd order differential equations in class, and have touched on superposition and were talking about an equation being linearly dependent or independent. I received some good explanations from tutors about this, using vectors as examples, but I'm still a bit unclear on the...

What is the history of the concept that a measurement process is associated with a linear opeartor? Did it come from something in classical physics? Taking the expected value of a random variable is a linear operator - is that part of the story?

A problem from matrix analysis. This proof seems to simple... is it correct? Problem 1: Suppose $T: V \rightarrow V$ is linear and $\mu \in F$. Show that if there exists a positive integer p such that $ker((T - \mu I)^p)$ $\ne \left\{ 0 \right\}$, then $\mu$ is an eigenvalue of T. Proof...