Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
The word linear comes from Latin linearis, "pertaining to or resembling a line".
I'm studying Differential Equations from Tenenbaum's, and currently going through non-homogeneous second order linear differential equations with constant coefficients. Method of Undetermined Coefficients is the concerned topic here. I will put forth my doubt through an example. Let's say we are...
Hi Everybody,
I am having some difficulties on the prove this problem.
I picked a nice example when I was trying to think about the proof.
Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...
Hello there, I have derived the expressions for electric field and potential to be the ones above, then for continuity at ##x = 0## I set the electric fields and potentials to be equal to yield the expressions:
$$Sx_p^2 = Kx_n^2$$
$$V_{bi} = V_n - V_p = \frac {q}{3\epsilon} \left( Sx_p^3 +...
for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection.
As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...
Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$}
https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing
ok I think this is the the definition we need for this practice exam question...
$\tiny{115.C51}$
find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$
$S=\left\{
\left[\begin{array}{r}2\\-1\\2\end{array}\right],
\left[\begin{array}{r}3\\0\\1\end{array}\right],
\left[\begin{array}{r}1\\1\\-1\end{array}\right]...
$\tiny{311.1.7.11}$
ok I am going to do several of these till I get it...
Find the value(s) of h for which the vectors are linearly dependent. Justify
$\left[\begin{array}{rrrrrr}
2\\-2\\4
\end{array}\right],
\left[\begin{array}{rrrrrr}
4\\-6\\7
\end{array}\right],
\left[\begin{array}{rrrrrr}...
I was wondering about EM waves produced by linearly accelerating charges, as opposed to oscillating charges.
With oscillating charges, the frequency of the wave depends on the frequency of the oscillation of the charge. But what determines the frequency of the wave produced by a linearly...
Hey! 😊
Let $1\leq n\in \mathbb{N}$, let $\mathbb{K}$ be a field, $V$ a $\mathbb{K}$-vector space with $\dim_{\mathbb{K}}V=n$ and let $\phi:V\rightarrow V$ be linear.
- Let $0\neq v\in V$ and $1\leq m\in \mathbb{N}$, such that $\phi^{m-1}(v)\neq 0=\phi^m(v)$. Show that $\phi^0(v)=v, \phi (v)...
Hey! 😊
Let $\mathbb{K}$ a field and let $V$ a $\mathbb{K}$-vector space. Let $1\leq m, n\in \mathbb{N}$ and $n=\dim_{\mathbb{K}}V$. Let $v_1, \ldots , v_m\in V$ be linearly independent.
Let $\lambda_1, \ldots , \lambda_m, \mu_1, \ldots , \mu_m\in \mathbb{K}$ such that...
I have an equation that comes from an especific topic of cam mechanisms and it goes like this:
$$
2M[tan(B)-B] - \beta Ntan(B) - 2\pi\sqrt{1 - N^2} = 0 \ \ \ \ \ \ \ \ \ (1)
$$
For this it doesn't matter what each variable means.
I'm trying to create a 3x3 matrix with a determinant equal to...
Summary:: I attach a picture of the given problem below, just before my attempt to solve it.
We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##.
I don't know...
We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t-\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t) - \frac{1}{3} \varphi_3(t) = 0##.
However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I...
Hey! :o
We have that the vectrs $\vec{v},\vec{w}, \vec{u}$ are linearly independent.
I want to check if the pairs
$\vec{v}, \vec{v}+\vec{w}$
$\vec{v}+\vec{u}$, $\vec{w}+\vec{u}$
$\vec{v}+\vec{w}$, $\vec{v}-\vec{w}$
are linearly indeendent or not.
Since $\vec{v}, \vec{w}, \vec{u}$...
16.1 Show that $e^{2x}$, sin(2x) are linearly independent on $(-\infty,+\infty)$
https://www.physicsforums.com/attachments/9064
that was the example but...
\begin{align*}
w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\
&=??\\
&=??
\end{align*}
Are the vectors
$$v_1=x^2+1
,\quad v_2=x+2
,\quad v_3=x^2+2x$$
linearly dependent or linearly independent?
if
$$c_1(x^2+1)+c_2(x+2)+c_3(x^2+2x)=0$$
is the system
$$\begin{array}{rrrrr}
&c_1 & &c_3 = &0\\
& &c_2 &2c_3= &0\\
&c_1 &2c_2& = &0
\end{array}$$
I presume at this point observation can...
Homework Statement
I'm trying to do a problem two ways, and things aren't consistent, finding the electric potential difference in a linearly varying field.
The electric potential difference between two points is often summarized in texts as ΔV = Vf - Vi = - ∫ E⋅ds where the lower bound of...
It is well known that the set of exponential functions
##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##,
with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions
##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##,
with ##k\in\mathbb{R}_+##.
What about...
i saw a proof that said “in R/{0} , the set [-1,0) has an upper bound ,but no least upper bound. no such set exists in linearly ordered R” ,but i could not understand it.
Homework Statement
An object of mass 50 kg gains 20,000,000 joules every second. Devise formulae to find the distance covered at any given point in time, and the time necessary to cover a certain distance.
Homework EquationsThe Attempt at a Solution
E = 20,000,000 * t
V = (E * 2/50)1/2
That's...
I need to scale my solar system model and it's not working. I'm making an augmented reality app that shows a sample solar system rotating. It works great! (Thanks to this $7 program https://assetstore.unity.com/packages/tools/physics/gravity-and-orbits-solar-system-105300 . The Unity engine...
Homework Statement
A positron is moving in a circular orbit of radius r = 2cm within a uniform magnetic field B0 = 50##\mu##T. The magnetic field varies over time according to the expression:
B = 700t + Bo
and, therefore, each orbit can be considered almost circular.
(a) Calculate the...
Homework Statement
Hi, I am having trouble understanding how the B1 field as described by (3.48) in the image attached in MRI which is described as a linearly polarized field is decomposed into it's final two circularly polarized field as described by (3.49) in the image attached.
Homework...
Solved (sorry i tried again and realized my E-values were wrong)
1. Homework Statement
Homework Equations
Find Eigen Values and then what?
The Attempt at a Solution
I got eigen values as 3 and -3.
Now how to proceed?
I got Eigen Vector as: 1, 1 for eigen value of 3
and eigen vector as 8, 2...
Homework Statement
Given the six vectors below:
1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation.
2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...
Homework Statement
[/B]
##X## is linearly independent if and only if every nonzero element of the subgroup ##\langle X \rangle## may be written uniquely in the form ##n_1 x_1 + ... n_k x_k## (##n_i \in \Bbb{Z} \setminus \{0\}##, and ##x_1,...,x_k \in X## are distinct).
Homework Equations
[/B]...
I'm trying to design something and rather than solve it using FEA guess and check, I'm trying to mathematically solve it.
The part I'm getting stuck at is having 2 point loads on a linearly varying cantilever beam.
https://imgur.com/LtHCvCB
The closest example I could find to this that was...
Suppose we had an infinite series -
z = ∑i = 1 to ∞ ( α1(i)x1 + α2(i)x2 + . . . + αm(i)xm )
- rewritten as the cumulative sequence -
z(n) = α1(n)x1 + α2(n)x2 + . . . + αm(n)xm
- where the xj are linearly independent and normalized (and serve as a finite basis across the sequence). If all...
The Friedmann equation for a spatially flat Universe is given by
$$\Big(\frac{\dot R}{R}\Big)^2=\frac{8 \pi G}{3}\rho$$
where ##R(t)## is the proper radius of some spherical volume with us at its center.
Let us assume that there is a mass ##M## inside this spherical volume of radius ##R##. The...
I have two questions for you.
Typically when trying to find out if a set of vectors is linearly independent i put the vectors into a matrix and do RREF and based on that i can tell if the set of vectors is linearly independent. If there is no zero rows in the RREF i can say that the vectors are...
Homework Statement
[/B]
Give examples of fixed points of vector fields and maps that are stable in the linear approximation but are nonlinearly unstable
Homework EquationsThe Attempt at a Solution
[/B]
I was able to find an example in a vector field that, when the Jacobian is found and the...
Homework Statement
If ##r_1, r_2, r_3## are distinct real numbers, show that ##e^{r_1t}, e^{r_2t}, e^{r_3t}## are linearly independent.
Homework EquationsThe Attempt at a Solution
By book starts off by assuming that the functions are linearly dependent, towards contradiction. So ##c_1e^{r_1t}...
(sorry for the horrible butchered thread title... should say "determination", not "determining")
1. Homework Statement
In "Principles of Quantum Mechanics", by R. Shankar, a vector space is defined as having dimension n if it can accommodate a maximum of n linearly independent vectors (here is...
Consider a spherical raindrop that falls at a constant velocity and whose radius ##r## is proportional to the distance ##h## fallen, i.e., ##r=kh##. Find the loss of gravitational potential energy (GPE) after it has fallen a distance ##h##.
The given answer is ##mg\frac{h}{4}## but my answer is...
How can I find out if this matrix A's columns are linearly independent?
$\begin{bmatrix}1&0\\0&0\end{bmatrix}$
I see here that $x_1 = 0$ and similarly $x_2 = 0$ does this mean that this matrix A's columns are therefore linearly dependent?
Also this is a projection onto the $x_1$ axis so is it...
Homework Statement
Here is my problem : I want to know where does group delay equation come form?
Homework Equations
As I checked reference 21 it only indicates
The Attempt at a Solution
I'm following how can I start and solve this problem
In the uploaded file, question 11 says that in b) the solutions y1 and y2 are linearly independent but the Wronskian equals 0. I think it said that they are independent because it's not a fixed constant times the other solution (-1 for -1<=t<=0 and 1 for 0<=t<=1) but it clearly says in the...
Homework Statement
True or false.
If v1... v4 are linearly independent vectors in 4D space, then {v1,v2,v3} is also linearly independent.
There's a hint: Think about (c for different constant) cv1+cv2+cv3+0*v4=0
I know that linear independence means there's only the trivial solution... which...
Homework Statement
why the vertical rise of the surface is given by the formula as shown in the notes ? shouldn't it = Δs =
(g +az ) Δx / ax ?
Homework EquationsThe Attempt at a Solution
Hey all, student100's brother here, he got me to create an account to ask my question here...
I'm taking an introduction to linear algebra class and we had a test problem to prove a set of vectors is linear independent: ##V = v_1, \dots, v_n \in \Bbb R^n## such that each element of the set is...
Homework Statement
A cart roles down the track with an initial velocity vo. Because of falling rain, water starts filling the cart such that its mass increases linearly with time. The rain that has fallen on the track cause the wagon to experience a frictional force characterized with a...
Homework Statement
Prove that a set of linearly independent vectors in Rn can have maximum n elements.
So how would you prove that the maximum number of independent vectors in Rn is n?I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of the...
why we assume that velocity decreases linearly in a spring (i.e. if one end is fixed, then velocity of a particle (of spring) at x from fixed =vx/l where v is the velocity of the free end) and why does it hold good too when the spring (linear mass density) is non uniform...
EDIT: spring has...
Hi,
Imagine a conductive wire bent to the shape of a loop without its ends meeting. A magnet is moved with respect to the loop such that the magnetic field crossing it (perpendicularly) is linearly increasing with time (Φ=kt) where k is a constant. The induced emf is the rate of change of...
I was wondering if time on a planet which is exactly twice the mass of the Earth would pass exactly twice as slow relative to Earth time?
So after a year on this big earth, two years would have passed on earth?
Thanks
Homework Statement
A linear transformation with Matrix A = ##
\begin{pmatrix}
5&4&2\\
4&5&2\\
2&2&2
\end{pmatrix} ## has eigenvalues 1 and 10. Find two linearly independent eigenvectors corresponding to the eigenvalue 1.
Homework Equations
3. The Attempt at a Solution [/B]
I know from the...
Homework Statement
Solve 2x''+3x'+40x = 40y+3y'
Homework Equations
y = 0.05sin(10t)
The Attempt at a Solution
I used the annihilator method to find the answer of x(t) = Acos(10t)+Bsin(10t)+Ce-0.75tcos(sqrt(311)/4t)+De-0.75tsin(sqrt(311)/4t) where A, B, C and D are constants.
The initial...
Hi guys, there are a lot of textbooks and articles describe the electric field in depletion region at thermal equilibium, but very few of them tell what happens to the field under bias. I am wondering if the electric field in the depletion region simply changes linearly with reverse bias? Or if...
[Note: no template because this post was moved from the QM subforum]
I was working on problem #41 and was confused about what the wave function would look like from the time x = 0 to when E=V0. (See image in...