Linear Definition and 1000 Threads

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...

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...

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...

Homework Statement I did an experiment on how stretching a rubber band, affects the range or distance of a projectile shot. Variable list: Independent Variable Amount of stretching Dependent Variable Range or distance I created my catapults on my own and used two different rubbers bands...

Homework Statement If ##A## is an ##n \times n## matrix, show that the eigenvalues of ##T(A) = A^{t}## are ##\lambda = \pm 1## Homework EquationsThe Attempt at a Solution First I assume that a matrix ##M## is an eigenvector of ##T##. So ##T(M) = \lambda M## for some ##\lambda \in \mathbb{R}##...

Homework Statement T/F: Let ##T: V \rightarrow W##. If ##\{v_1,v_2,...,v_k \}## is a linearly independent set, then ##\{T(v_1), T(v_2),..., T(v_k) \}## is linearly independent. Homework EquationsThe Attempt at a Solution This seems to be true, because we know that ##a_1v_1 + a_2v_2 + \cdots +...

The function ƒ(x) is a linear function and g(x) is a rational function. These functions have the following values: ƒ(3) = 7 g(3) = 5.6 ƒ(4) = 5 g(4) = 6.7 There is a solution to the equation ƒ(x) = g(x) between x = 3 and x = 4 that must be closer to 3 than 4. TRUE or FALSE?

Can i Use a standard F-test on longitudinal data for a linear multiple regression? Mons

Homework Statement http://prntscr.com/eqhh2p http://prntscr.com/eqhhcg Homework EquationsThe Attempt at a Solution I don't even know what these are, it is not outlined in my textbook. I'm assuming I am is image? But how do you calculate image even? As far as I'm concerned I am has to do wtih...

Say I want to see if a continuous variable, say strength, is associated with another continuous variable, say height. Clearly I can just use linear regression with height as the response variable and strength as the predictor variable. But I can also split height into a tall category and a...

Hi, I'm currently doing my extended essay for the IB in physics and I'm basing mine around the Gauss gun (similar post that explains it really well here https://www.physicsforums.com/threads/physics-behind-gaussian-accelerator-magnetic-linear-accelerator.320389/). What I was wondering was how I...

Hey. I am doing a project where I am studying a set of companies over a 7-year period. I am doing a multiple linear regression analysis either with fixed or random effects (so, it's a panel study). What I am wondering is if the general assumptions/requirements apply when using the fixed/random...

Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear...

Can i have help with this linear differential equation ? First, i divided by (1-x^2) to be like dy/dx + p(x)y= q(x). But i could not obtain Q(x). Any help will be welcomed.

Homework Statement A 500g putty ball moving horizontally at 6m/s collides with and sticks to a block lying on a friction-less horizontal surface. If 25% of the kinetic energy is lost, what is the mass of the block? Homework Equations initial (i) = final (f) m1v1+m2v2 = m1v1+m2v2...

Homework Statement Find the general solution of y^{(5)}-y(1)=x The Attempt at a Solution I found the complementary function by substitution of the solution form y=e^{kx} giving k=0,1,-1,i,-i, so y_{cf}=a_0+a_1e^x+a_2e^{-x}+a_3e^{ix}+a_4e^{-ix} Now for the particular integral, the general...

Hi the parameter $\lambda$ in linear integral equations say Fredholm that appears in front of the integral containing the kernel i.e. $y(x)=f(x)+ \lambda \int_{a}^{b} \,k(x,t) y(t) dt $ can $\lambda$ be adjusted for the convergence of the method like in using Picard's successive...

Homework Statement [/B] A uniform cylinder with radius r is rolling down an incline without slipping as shown in diagram below. It's angular acceleration is α and acceleration of it's center of mass C is a. What will be the linear acceleration of point of contact P and another point Q on the...

I have been studying structural physics lately, in particular the linear elastic eqns, governed by the equation of motion. https://en.wikipedia.org/wiki/Linear_elasticity I am trying to solve these equations numerically. I was thinking of a scenario of a rod being compressed on one end by some...

I'm in a bit of a dilemma. At my university, Calc 2 is a prerequisite to Linear Algebra. However, I have been told that it's totally doable to take both at the same time. What do you guys think? Do most universities wave this requirement if you take them concurrently? Thanks!