Linear Definition and 1000 Threads

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...

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?

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...

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]...

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...

This is an exercise given in my real analysis course. We have definitions for continuity, uniform continuity, l.s.c and u.s.c., but I don't see how to apply them. I guess I want to construct $φ$ piece by piece of the interval to eventually cover all of [a,b]? Exercise: A continuous function...

What I have done: I changed all fractions to common denom and that gave me 5y-5x=1 (1) *I numbered the fractions 5y+2x=5 (2) Then: 5y=5-2x Substitute into equation 1 (5-2x)-5x=1 5-7x=1 x=4/7 Thing is my answer says I should be getting x=0 Any hints?

Hello, We defined a k-form on a smooth manifold M as a transfromation Where the right space is the one of the alternating k-linear forms over the tangent space in p. If we suppose we know, that we get a basis of this space by using the wedge-product and a basis of the dual space, then we might...

Problem: Suppose V is a complex vector space of dimension n, and T is a linear map from V to V. Suppose $x \in V$, and p is a positive integer such that $T^p(x)=0$ but $T^{p-1}(x)\ne0$. Show that $x, Tx, T^2x, ... , T^{p-1}x$ are linearly independent.During class my professor said it was "a...

[solved] Basis for set of solutions for linear equation Hi, I have this problem I was working through, but I'm not sure that I've approached it from the right way. The problem consists of 3 parts, which build off of each other. I'm pretty confident about the first two parts, but no so much...

I have a question about HHL algorithm https://arxiv.org/pdf/0811.3171.pdf for solving linear equations of the form: A x = b Where A, x and b are matrices Take for example 4x1 + 2x2 =14 5x1 + 3x2 = 19 HHL apply the momentum operator eiAτto/T on the state, do a Fourier Transform on |b> and...

Hey all, I don't understand what makes a differential equation (DE) linear. I found this: "x y' = 1 is non-linear because y' is not multiplied by a constant" but then also this: "x' + (t^2)x = 0 is linear in x". t^2 also isn't a constant. So why is this equation linear?

I'm a physics student who has the option to take some advanced math courses (Real analysis through Rudin and beyond, functional analysis if I have time, as well as algebra through Artin). I'm only just going into my second year this term, and will either be retaking linear algebra 2, or taking...

Homework Statement Homework Equations 3. The Attempt at a Solution [/B] ## |3 \rangle = |1 \rangle - 2 ~ |2 \rangle ## So, they are not linearly independent. One way to find the coefficients is : ## |3 \rangle = a~ |1 \rangle +b~ |2 \rangle ## ...(1) And solve (1) to get the values of a...

I purchased a buck regulator I apply 12 volts dc to in and get 11.5 volts out turn screw on pot makes no changes measured pot goes from 4k to zero ohms if one unhooks connection to terminal 1 adjustment to LM317 will this give you the minimum voltage of about 1.5-1.6 volts? I want 12 volts in...

I'm analysing some data and my task is to get a line that best fits the data, using least square I'm getting these dashed curves (red and blue) with low correlation factors. Is there another method that takes into consideration the amount of data placed into the direction of a line?