Linear Definition and 1000 Threads

Homework Statement If A is an n x n matrix over R such that A^3 = A + 1, prove that det(A) > 0 . Homework EquationsThe Attempt at a Solution So, what I've done is factor the expression to get A(A+1)(A-1) = 1, then taking the determinant of both sides, I get det(A)det(A+1)det(A-1) = 1. I...

Homework Statement Given $$u''(x)+\lambda u = 0\\ u(-1)=u(1)=0.$$ If ##\lambda_0## is the lowest eigenvalue, show that ##4 \lambda_0 = \pi^2##. Homework Equations $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}$$ where ##glb## denotes greatest lower bound and ##L## is the Sturm-Louiville operator. I...

Homework Statement Hello to everyone that's reading this. :) For this linear least-squares regression problem (typed below and also), I correctly find the value of g (which is what the problem statement wants to have found), but I was curious about the value of ##a_0## (and that's what this...

Homework Statement "Show that every subspace of ##ℝ^n## is the set of solutions to a homogeneous system of linear equations. (Hint: If a subspace ##W## consists of only the zero vector or is all of ##ℝ^n##, ##W## is the set of solutions to ##IX=0## or ##0_vX=0##, respectively. Assume ##W## is...

Homework Statement "Find ##S_\alpha## where ##S: M_{2×2}(ℝ)→M_{2×2}(ℝ)## is defined by ##S(A)=A^T##. Homework Equations ##A^T=\begin{pmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{pmatrix}## ##\alpha= \{ {\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0...

Homework Statement "Determine whether the function ##T:M_{2×2}(ℝ)→ℝ## defined by ##T(A)=det(A)## is a linear transformation. Homework Equations ##det(A)=\sum_{i=1}^n a_{ij}C_{ij}## The Attempt at a Solution I'm assuming that it isn't a linear transformation because ##det(A+B)≠det(A)+det(B)##...

Hey all, I'm currently working on my CS degree with a mathematics minor. After this Fall, I will only have one more course to take to finish my minor. I'm debating between Linear Algebra and Deterministic Operations Research. I do have other options, but these seem to be most applicable to CS...

I have some data that I want to do simple linear regression on. However I don't have a lot of datapoints, and would like to know the uncertainty in the parameters. I.e. the slope and the intercept of the linear regression model. I know it should be possible to get a prob. distribution of the...

Homework Statement Use definition (1) to determine if the functions ##y_1## and ##y_2## are linearly dependent on the interval (0,1). ##y_1(t)=cos(t)sin(t)## ##y_2(t)=sin(t)## Homework Equations (1) A pair of functions is said to be linearly independent on the interval ##I## if and only if...

Homework Statement Linear Algebra Problem: Solving for Euler between two ordered bases I've got a problem I need to solve, but I can't find a clean solution. Let me see if I can outline the problem somewhat clearly. Okay, all of this will be in 3D space. In this space, we can define some...

Cross-posted on SE.DS Beta. I'm just doing a simple linear regression with gradient descent in the multivariate case. Feature normalization/scaling is a standard pre-processing step in this situation, so I take my original feature matrix $X$, organized with features in columns and samples in...

Imagine that you own a grove of orange trees, and suppose that from past experiences you know that when 100 trees are planted, each tree will yield about 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases...

Hi all, I'm stuck on progressing a problem i have received some feedback around as detailed below. I would greatly appreciate some assistance, and thank you in advance for your time and contributions. So i have a linear system: Which is row reduced to: I have identified that the system has...

Homework Statement Consider a straight line segment of 3L and with a linear charge density λ. Determine the electric field, E, of at point P, which is a point within the segment and along the axis. (figure attached) Homework Equations dE=kdQ/r^2 The Attempt at a Solution I attempted solving...

Homework Statement Let ##T:M_2 \to M_2## a linear transformation defined by ##T \begin{bmatrix} a&b\\ c&d \end{bmatrix} = \begin{bmatrix} a&0\\ 0&d \end{bmatrix}## Describe ##ker(T)## and ##range(T)##, and find their basis. Homework Equations For a linear transformation ##T:V\to W##...

Hey everyone! (new to the forum) I am currently trying to self study more advanced mathematics. I have taken up to multivariable calculus and have taken a class for an introduction to mathematical proofs/logic (sets, relations, functions, cardinality). I want to get a head start on the...

Homework Statement Let T: ℝ² → P² a linear transformation with usual operations such as T [1 1] = 1 - 2x and T [3 -1]= x+2x² Find T [-7 9] and T [a b] **Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper...

Homework Statement Let be T : ℙ2 → ℙ2 a polynomial transformation (degree 2) Defined as T(a+bx+cx²) = (a+1) + (b+1)x + (b+1)x² It is a linear transformation? Homework Equations A transformation is linear if T(p1 + p2) = T(p1) + T(p2) And T(cp1)= cT(p1) for any scalar c The Attempt at...

The term simultaneous in simultaneous linear equations does not make sense to me? Would you explain the what simultaneous mean there? Example: "We have all solved simultaneous linear equations - for example, 2x + y = 4 x - 2y = -3 " Source: Linear Algebra by Fraleigh/Beauregard. Thank you.

What does "linear" in linear algebra and "abstract" in abstract algebra stands for ? Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##...

Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## Homework EquationsThe Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int e^x...

Homework Statement q=1.602*10^-19 point 1 L=1mm=r1 v=1.1*10^6 at point 2 F=1.44*10^-12 at point 1 Homework Equations E=(1/4πε)*(q/r) ΔV=∫E*dr=(1/4πε)*q∫(1/r)=(1/4πε)*q*ln (r2/r1) ΔU=ΔK=mv^2/2 ΔK=mv^2/2=ΔV*q=q*(1/4πε)*Q*(ln(r2/r1))...

Homework Statement Hi everybody! In my quantum mechanics introductory course we were given an exercise about the 3D quantum harmonic oscillator. We are supposed to write the state ##l=2##, ##m=2## with energy ##E=\frac{7}{2}\hbar \omega## as a linear combination of Cartesian states...

Homework Statement http://i.imgur.com/swYr8aw.jpg Homework Equations delta L=LalphadeltaT The Attempt at a Solution sorry for s**ty handwriting look at #8[/B] http://i.imgur.com/59Ew8vJ.jpg edit:click the links please. The forum software cropped the quality of the photo.

Homework Statement Let { u, v, w} be a set of vectors linearly independent on a vector space V - Is { u-v, v-w, u-w} linearly dependent or independent? Homework Equations [/B] A set of vectors u, v, w are linearly independent if for the equation au + bv + cw= 0 (where a, b, c are real...

Hello, I know how a plunger in a pull solenoid is pulled to the center of the solenoid. What I am wondering is the following. Does the plunger experience a centering force towards the center of the coil? If the plunger is moved slightly off-axis from the center of a circular coil, will the...

I am trying to derive the DC electrical conductivity using the pertubation theory in Interaction picture and linear response theory. If working in a energy eigen basis and using the density matrix, the Fourier transform of the susceptibility can be written as ##\chi {(\omega )_{ij}} =...

I'm reading a book on vortex methods and I came across the above mentioned terms, however, I don't understand what they mean in mathematical terms. The book seems to be quite valuable with its content and therefore I would like to understand what the author is trying to say using the above...

During a lab exercise we measured different masses of a magnetic material on a scale while changing the strength of the magnetic field it was in. Afterwards we plotted the masses and the fieldstrength hoping to find a linear slope. Then we drew a linear slope by using linear regression and found...

In the course of another thread I was lend to think about the Raman effect. I also read about the stimulated Raman effect and found that it is usually described as a third order nonlinear effect where a power of two of E is assumed to drive the nuclear vibration. I don't quite see why this is...

The Friedmann equation expressed in natural units (##\hbar=c=1##) is given by $$\left(\frac{\dot a}{a}\right)^2 = \frac{l_P^2}{3}\rho - \frac{k}{R^2}$$ where ##t## is the proper time measured by a comoving observer, ##a(t)## is the dimensionless scale factor, ##l_P=\sqrt{8\pi G\hbar/c^3}## is...

I had a pretty tough schedule this semester, so I'm getting my first B. I otherwise wouldn't be too sad, but I hear Linear is pretty important in upper-level physics and astronomy. So, will this hurt my chances of getting into grad school? I am (was? rising sophomore) only a first-year, and I do...

Homework Statement Let V be a vector space over R. let Φ1, Φ2 ∈ V* (the duel space) and suppose σ:V→R, defined by σ(v)=Φ1(v)Φ2(v), also belongs to V*. Show that either Φ1 = 0 or Φ2 = 0. Homework Equations N/A The Attempt at a Solution Since σ is also an element of the duel space, it is...

I've managed to derive the form of Reynolds transport theorem as a bilance of linear momentum of the system: \left (\frac{\vec{\mathrm{d} p}}{\mathrm{d} \tau} \right )_{system}=\frac{\mathrm{d} }{\mathrm{d} x}(\int_{V}^{ }\vec{v}\cdot \rho dV)+\int_{A}^{ }\vec{a}dm+\int_{A}^{ }\vec{v}\cdot \rho...

Homework Statement Hi guys, I am having an issue understanding what to do with this question. The question is displayed below: I have hand wirtten my working, as I don't now how to do matrices fully on latext. I used the definition to get this far for part a, but not sure about the second...

I had this setup (see attached photo) for the linear gate method in a γγ coincidence experiment. Using a Na22 source. The pulse from the movable detector enables the gate of the MCB, and any corresponding pulse from the fixed detector that arrives within the gate interval will be considered...

Homework Statement Hi guys, I am having a bit of trouble with this question: S2. It the linear non linear and homogeneous parts. I think it is a linear equation, as I always think dy/dx (y)=H(x), but is there a way to show this, also for non linear cases. I believe the second part to this...

Suppose that the relation R is defined on the set Z where aRb means a = ±b. Establish whether R is an equivalence relation giving your justifications. Find the set of solutions of each of the linear congruence: a) x ≡ 3 (mod 5). b) 2x ≡ 5 (mod 9).(please write the full solutions thanks)

Is it true that torque and axial force are stronger than linear force? If I hit something with an acute angle and progressively turn it into a lower angle while moving forward,is it applying more force?( for example needle insertion)

Homework Statement Find the general solution to the equation: ##{u}\times{(i+4j)}=3k## Homework Equations ##u=(ai+bj+ck)## The Attempt at a Solution I am having trouble visualising what to do here. So my throught so far is that, if i do ##(ai+bj+ck)\times {(i+4j)}=(0i-0j+(4a-b)k) ## now if I...

Hi, I have 2x+3y=0 equation, I need to find x and y points to draw a line on plane. Can anyone help me?

I am wondering, when solving rigid body exercises, how can I express the relationship between linear and angular acceleration for a general case? E.g. what would be the linear acceleration in function of the angular one of a 1m rod that is rotating through a fixed point 0.6 m away from its mass...

I have a question about weights of a basis set with respect to the other basis set of one specific vector space. It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation" Let's consider a vector space...

Classical physics is a nonlinear theory, but how is it that? Why is it nonlinear? Also quantum mechanics is a linear theory so that the sum of the solutions of the schrödinger equation is itself a solution. But I'm not sure I grasp this completely. Why is quantum mechanics linear while...

Homework Statement 3.For which values of ##\lambda## does the following system of equations also have non trivial solutions Homework EquationsThe Attempt at a Solution What I tried doing first is to put all variables on the same side and got ## v+y-\lambda*x=0\\ x+z-\lambda*y=0\\...

Homework Statement Say I have a matrix: [3 -2 1] [1 -4 1] [1 1 0] Is this matrix onto? One to one? Homework EquationsThe Attempt at a Solution I know it's not one to one. In ker(T) there are non trivial solutions to the system. But since I've confirmed there is something in the ker(T), does...

So I am working out a course schedule for my last two years of undergrad and have room for only one more math class but do not know which would be more beneficial. The two courses are Intro to Modern Algebra or Numerical Linear Algebra. I am working towards a bachelors degree in physics and plan...

Homework Statement Consider the vector space that consists of all possible linear combinations of the following functions: $$1, sin (x), cos (x), (sin (x))^{2}, (cos x)^{2}, sin (2x), cos (2x)$$ What is the dimension of this space? Exhibit a possible set of basis vectors, and demonstrate that...

Homework Statement Suppose u,v ∈ V and that Φ(u)=0 implies Φ(v)=0 for all Φ ∈ V* (the duel space). Show that v=ku for some scalar k. Homework Equations N/A The Attempt at a Solution I've managed to solve the problem when V is of finite dimension by assuming u,v are linearly independent...

Homework Statement The vectors ##a_1, a_2, a_3, b_1, b_2, b_3## are given below $$\ a_1 = (3~ 2~ 1 ~0) ~~a_2 = (1~ 1~ 0~ 0) ~~ a_3 = (0~ 0~ 1~ 0)~~ b_1 = (3~ 2~ 0~ 2)~~ b_2 = (2 ~2~ 0~ 1)~~ b_3 = (1~ 1~ 0~ 1) $$ The subspace of ## \mathbb R^4 ## spanned by ##a_1, a_2, a_3## is denoted by...