Linear Definition and 1000 Threads

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?

Given a square with a linear mass density of: λ(x) = a * x (see image below where black is high density and white is low density) how would you deduce what the surface mass density is? I get confused for the following reason: To me it seems that the surface mass density should depend on x...

so say I suspect that there is a positive trend in the data from the scatter plot. Say the output y is continuous. A linear regression would give me a possitive estimate of the slope. For a one unit increase in x, I would get a so and so increase in y. I can also split the data for the y...

Homework Statement Suppose that AB = 0, where A is a 3 x 7 full rank matrix and B is 7 x 53. What is the highest possible rank of matrix B. Homework EquationsThe Attempt at a Solution Since each column of B is in the null space of A, the rank of B is at most 4. I don't understand why it is...

Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of...

According to my text, a linear system of equations is a problem described by two or more equations in two or more variables. Now the individual equations have infinitely many solutions, however, the system of equations is said to have either exactly one solution (one point of intersection...

$$\tiny{la1.1.22}$$ $\textsf{Construct 3 different augmented matrices for linear systems whose solution set is}$ $$x_1=3,\quad x_2=-2,\quad x_3=-1$$Well we could start just by$$3x-2y+y=-1$$but then we need a $3\times4$ matrix

Hello, could someone please give me some examples of where order linear non homogenous ordinary differential equations are used in physics[emoji4]

After seeing so much higher-level physics and proofing for special relativity, I imagine I'll need to take this at some point to continue do grad-level physics. I'm taking calc III at the start of year two, and then on to diff eq. When should I take linear algebra in that case? My adviser seemed...

Homework Statement So imagine 4 rigid rods connected together to form a dashed quadrilateral as shown in the picture. Now AB is fixed, can not be changed in anyway, while all other sides (AD, BC and CD) are connected but can move freely. The initial conditions (dashed quadrilateral) are given...

Homework Statement Assertion- If linear momentum of particle is constant, then its angular momentum about any axis will also remain constant Reason-Linear momentum remains constant when net force is 0, angular momentum remains constant when net torque is zero which of these statements is/are...

Homework Statement Solve the following system of equations Homework Equations A1 + A3 = 0 cA1 + A2 + A4 = 0 dA1 + cA2 + w^2 A3 = 1 dA2 + w^2A4 = 0 The Attempt at a Solution https://imgur.com/YSG5oGN - Page 1 https://imgur.com/dGbccsE - Page 2

Homework Statement In the follow cases find a maximal linearly independent subset of set ##A##: (a) ##A = \{(1,0,1,0),(1,1,1,1),(0,1,0,1),(2,0,-1,)\} \in \mathbb{R}^4## (b) ##A = \{x^2, x^2-x+1, 2x-2, 3\} \in \mathbb{k}[x]## The Attempt at a Solution The first part of the exercise is...

I have some experimental data, in this case, we performed a study of the Zeeman effect in Cadmium with the use of a Fabry-Perot inferferometer. The data should fit a straight line, but I would like to force the intercept through the origin since the relation between the wavenumber difference and...

Homework Statement The object of the figure below has 2 disks attached on both sides through and axel of negligible mass. Both disks of mass M. I need to find the linear acceleration of this object knowing there exists a tension T on the cord. Homework Equations :[/B] T=αI The Attempt at a...

Homework Statement Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace. (a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B## (b) Let ##C \subseteq V## be a...

If the uncertainty in the age of the Universe is ##\Delta t## then the Uncertainty Principle implies that it has an uncertainty in its energy ##\Delta E## given by $$\Delta E \ \Delta t \sim h.\tag{1}$$ If this energy fluctuation excites the zero-point electromagnetic field of the vacuum then a...

Hi, Just wanted to see if I understood the meaning of Generalized Linear Models: In the case of Standard ( "Non-generalized") Linear models, a dependent variable y is a linear function of a dependent variable x. In a Generalized Linear Model (GLM), a dependent variable y is linear in some...

Homework Statement Find the Green's function for $$f''(x) + \cos^2 a f(x) = 0;\\ \pm f'(x) + \cos a \cot a f(x)|_{x=x_0(a)}=0$$ where ##a## is a parameter and ##x_0## is defined as $$x_0(a) = \sec a\arcsin(\cos a)$$. Homework Equations Standard variation of parameters The Attempt at a...

Homework Statement Homework Statement (a) Let ##V## be an ##\mathbb R##-vector space and ##j : V \rightarrow V## a linear transformation such that ##j \circ j = id_V##. Now, let ##S = \{v \in V : j(v) = v\}## and ##A = \{v \in V : j(v) = -v\}## Prove that ##S## and ##A## are subspaces and...

Homework Statement 1. (a) Prove that the following is a linear transformation: ##\text{T} : \mathbb k[X]_n \rightarrow \mathbb k[X]_{n+1}## ##\text{T}(a_0 + a_1X + \ldots + a_nX^n) = a_0X + \frac{a_1}{2}X^2 + \ldots + \frac{a_n}{n+1}## ##\text{Find}## ##\text{Ker}(T)## and ##\text{Im}(T)##...

Hello, Obviously using a piston, you convert linear movement to circular. Or using a turbine. Other methods/mechanisms?

I have before me that ##\cos 2x + \sin 2x = \sqrt{2} \cos (2x - \pi / 4)##. Where does this expression on the right come from? I tried to look on the internet but I couldn't really articulate it well enough to find anything on it.

I was reading a paper about deterministic chaos, where the author had commented that non-linear systems can be affected by this. My question is whether some linear systems can also have chaotic behavior. It is possible? Thank you

Homework Statement I need to find the discrete time equivalent of the following system: \begin{bmatrix} \ddot{x} \\ \dot{x} \\ \ddot{\theta} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 4.2042857 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 105.1071428 \\ 0 & 0 & 1 & 0 \end{bmatrix}...

Homework Statement Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ##...

Homework Statement A beach ball is thrown upwards with initial speed v0. The drag force is given by Fd = −mαv for a drag coefficient α and speed v. (a) the height h reached (b) Show that the height may be expressed in terms of the final speed when the ball strikes the ground vf (c)Finally show...

Hello everyone .I cannot understand what Q14 is asking about. May I know how to solve Q14 shown in the pic？

Hello everyone, I've always had this question in my mind: Can we convert the non-linear function into a system of linear functions? I don't know if this is actually something exist in math (I searched a little bit to be honest), but I'm really interested in this question because it would make...

I wanted to go through Calculus and then Linear Algebra following either of two paths: a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra OR b) Simmons Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol...

a) y'' + 3y' - y = 3sin3x This is not homogeneous. b) y'' + 3y' - y = 0 This is homogeneous. I see b) is homogeneous because it equals to 0. What are further conclusions for that. How we can predict particular solution in a) to be: y = Asin3x + Bcos3x? And how to predict solutions for other...

Homework Statement Find the distance from point P (1,7,3) to the line (x,y,z) = (-2,1,4) + s(1,-3,4), s is a free variable Homework Equations projnQP = ( QP⋅n/(lengthQP)(lengthn) )(n) The Attempt at a Solution I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm...

Homework Statement Find the cosine of the angle determined by the intersecting lines m1: (x,y,z) = (-2,1,4)+s(1,-3,4) m2: (x,y,z) = (-2,1,4)+t(-2,3,2) s and t are free variables Homework Equations a⋅b = length(a)length(b) cosθ The Attempt at a Solution I just did this equation using no...

Hi EVERYBODY: General knowledge: The homogeneous linear Fredholm integral equation $\mu\ \varPsi(x)=\int_{a}^{b} \,k(x,s) \varPsi(s) ds$ (1) has a nontrivial solution if and only if $\mu$ is an eigenvalue of the integral operator $K$. By multiplying (1) by $k(x,s)$ and...

Homework Statement Let A and B be nxn matrices with Eigen values λ and μ, respectively. a) Give an example to show that λ+μ doesn't have to be an Eigen value of A+B b) Give an example to show that λμ doesn't have to be an Eigen value of AB Homework Equations det(λI - A)=0 The Attempt at a...

Homework Statement Show that T:C[a,b] -> defined by T(f) = ∫(from a to b) f(x)dx is a linear transformation. Homework Equations Definition of a linear transformation: A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique...

Homework Statement Homework Equations None. The Attempt at a Solution I know that the standard matrix of a counterclockwise rotation by 45 degrees is: [cos 45 -sin 45] [sin 45 cos 45] =[sqrt(2)/2 -sqrt(2)/2] [sqrt(2)/2 sqrt(2)/2] But the problem says "followed by a projection onto the line...

Homework Statement If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction? I thought for sure yes, but the way I particular question I just...

Homework Statement For a single mechanical unit lung, assume that the relationship among pressure, volume, and number of moles of ideal gas in the ling is given by PA((VL)/(NL)a = K, where a = 1 and K is a constant. Derive the lowest-order (linear approximation to the relationship among changes...

Homework Statement Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by...