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".
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
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
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...
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
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'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...
Homework Statement
My son (aged 13) received his math test back, and got the following problem wrong:
Which of the following functions is linear?
A: f(x) = -3x + 5
B: g(x) = x^2
C: h(x) = 6x - (1/2)
D: p(x) = 4x/5
The attempt at a solution
He answered B, which is clearly wrong, but it wasn't...
Homework Statement
I need some help with a question on my assignment. It asks to set up a matrix from the linear equations, y=25x+70 and y=35x+40.
Homework Equations
How do I set this matrix up?
The Attempt at a Solution
I think that I have to rewrite it as 25x-y=-70 and 35x-y=-40. But then I...
Homework Statement
Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.
Homework Equations
f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)
The Attempt at a Solution
I...
I would like to compare four slopes I got from linear fits through five data points for each fit. The fitting results (for the slopes) are as follows:
1: 0.08885 ± 0.00991
2: 0.08744 ± 0.0118
3: 0.10288 ± 0.00669
4: 0.0926 ± 0.01285
My hypothesis is that they are all not significantly different...
I'm reading about the Simplex method to solve a Linear Program. If a program is rejected as being infeasible, is there a method to identify which equations are causing it to be infeasible and a technique for reducing the program in an optimal way? (I'm not sure what 'optimal' is exactly, but my...
Homework Statement
Consider a rocket subject to a linear resistive force, $$f = -bv$$, but no other external forces. Use Equation (3.29) in Problem 3.11 to show that if the rocket starts from rest and ejects mass at a constant rate $$k = -\dot{m}$$, then its speed is given by:
$$v =...
Homework Statement
Let ##X_1 \sim N(3,2^2)## and ##X_2 \sim N(-8,5^2)## be independent. Let ##U=aX_1+bX_2##. What is the distribution of ##U##
Homework EquationsThe Attempt at a Solution
As they are independent, we can write the distribution of ##U## as the convolution of the 2. So I get...
Working on a research project to develop a test standard for abrasion testing pressure washer hoses. Ideally this could be used on any sort of hose. We have a set procedure that has worked on a couple hoses that looks like this:
.Weight applied is constant
.Cycle speed is constant
.Constant...
Hello!
I have a question that maybe has to do more with Mathematics, but if you do experimental physics you find it quite often.
Let's assume that we want to measure two quantities x and y that we know that they relate to each other linearly. So we have a set of data points xi and yi...
Homework Statement
Homework Equations
Power series
ODE
The Attempt at a Solution
[/B]
Sorry for not typing all those things out from my phone..
How can I get C1?
And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...
Homework Statement
Homework Equations
T = 2pi * sqrt(m/k)
mv =m2v2 (LM)[/B]The Attempt at a Solution
[/B]
So T2 depends on the mass and not velocity. So i can find T2 = 2pi * sqrt([m/2]/k)
For A2 , i know that the amplitude before any collision is 1/2m1v1^2 = 1/2kA1^2
so solving that, i...
HI .I'm trying to prove that, for a linear transformation, it is worth that:
f(a\bar{x}+b\bar{y})=af(\bar{x})+bf(\bar{y}) for every real numbers a and b.
Until now, I have proved by myself that
f(\bar{x}+\bar{y})=f(\bar{x})+f(\bar{y}).
and , using this result i proved that:
f(a\bar{v}) =...
Homework Statement
Homework EquationsThe Attempt at a Solution
I understand the solution of the problem, but I want to know why don't my approach work.
I didn't change the frame of reference.
I use the regular method of solving a torque problem.
(mgsinθ - mAcosθ)(L/2) = mL2/3 θ''
which is...
Hy folks,
Upfront I want to apologize for my writing and my dissability to use correct symbols to ease readability of the example. Ok now that that's done I just want to start upfront.
If we set a usual example of an object falling from a tower with a height of x meters and assume that the...