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".
Hi; I am missing something. I can follow the technicality of a homogenous linear equation has all coefficients of zero and the "contra" for non homogenous equations. I just can't figure out the relevance of the consequences of outcome. If I am not being clear maybe I can be guided as to how...
Hello Forum,
I have read about an interesting example of multiple linear regression (https://online.stat.psu.edu/stat501/lesson/12/12.3). There are two highly correlated predictors, ##X_1## as territory population and ##X_2## as per capita income with Sales as the ##Y## variable. My...
The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
I'm guessing this question can be solved using the law of conservation of momentum
Vi = 5 m/s
(5 m/s) M = (4.33 m/s) cos30 M + V sinθ M
I don't know what to do after this... I'm also not sure if I use the sin and cos correctly.
Hello,
I have been pondering on the following: we have data for blood pressure BP (response variable) and data about age and gender (categorical variable with two levels). We can build two linear regression models: $$BP=b_0+b_1 age+b_2 gender$$ $$BP=b_0+b_1 age$$
The first model does not take...
Dear everybody,
I am having some trouble proving the implication (or the forward direction.) Here is my work:
Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...
I am trying to find Planck's constant using Excel given the data:
Frequency [Hz]
Photon Energy [J]
7.5E+14
4.90E-19
6.7E+14
4.50E-19
6E+14
4.00E-19
5.5E+14
3.60E-19
5E+14
3.30E-19
4.6E+14
3.00E-19
4.3E+14
2.80E-19
4E+14
2.65E-19
3.75E+14
2.50E-19
I am using Linear...
Imagine two equal masses, m, moving through flat spacetime with opposite 3-momenta, as seen from an inertial frame in the COM.
In the massless case of two parallel, non-colinearly infinitely long moving bundles of light, Bonnor Beams, the beams are curved if the momenta are opposite, and stay...
Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.
Hello,
In studying linear regression more deeply, I learned that scaling play an important role in multiple ways:
a) the range of the independent variables ##X## affects the values of the regression coefficients. For example, a predictor variable ##X## with a large range typically get assigned...
Hello forum,
I have created some linear regression models based on a simple dataset with 4 variables (columns). The first models simply involve one predictor variable: $$Y=\beta_1 X_1+\beta_0$$ and $$Y=\beta_2 X_2+ \beta_0$$
The 3rd model is multiple linear regression model involving the 3...
For this problem,
Where equation 16.27 is the wave equation.
The solution is
I don't understand how they got the second partial derivative of ##y## with respect to
##x## circled in red.
I thought it would be ##1## since ##v## and ##t## are constants
Many thanks!
If I have been given a system of inhomogeneous linear ODEs,
$$
\vec{x'} =
\begin{bmatrix}
4 & -1 \\
5 & -2 \\
\end{bmatrix}
\vec{x}
+
\begin{bmatrix}
18e^{2t} \\
30e^{2t}\\
\end{bmatrix}
$$
I have found its particular solution to be:
$$
1/4
\begin{bmatrix}
-31e^{2t} - 25e^{6t} \\
85e^{2t} -...
Hello,
I have a question about linear regression models and correlation. My understanding is that our finite set of data ##(x,y)## represents a random sample from a much larger population. Each pair is an observation in the sample.
We find, using OLS, the best fit line and its coefficients and...
For part a:
I know that linear charge density is the amount of charge per unit length, and we are given the volume charge density. Since we are given the volume, we can obtain the length by multiplying the volume by the cross sectional area, so C/m^3 * m^2 = C/m. The cross sectional area of a...
Dear,
I would like to create a controlled system to inject 1-2 ml of liquid (a chemical reagent) into a solution contained in a glass flask.
The only request made is to use a reliable linear piston of minimum size, equal to or slightly larger than the injected volume (1 cm3) so that it can...
Hello,
Simple linear regression aims at finding the slope and intercept of the best-fit line to for a pair of ##X## and ##Y## variables.
In general, the optimal intercept and slope are found using OLS. However, I learned that "O" means ordinary and there are other types of least square...
I'm studying "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Stephen Newman. Theorem 4.4.4 in that book:
The proof of part 2 is given like this:
Seems a bit incomplete. I'd like to know if my approach is correct:
$$\langle A(v+tw),A(v+tw)\rangle=\langle...
How do we have linear spring direction (mostly a spherical spring) to have pull/push force evenly across some points within a range?
Or is it possible to create spring material with anomaly property capable of performing so?
I'm watching a nice video that tries to explain how linear algebra enters the picture in quantum physics. A quick summary:
Classical physics requires that physical quantities are single-valued and vary smoothly as they evolve in time. So a natural way to model classical physical quantities is...
A quasar with a bolometric flux of approximately 10−12 erg s−1 cm−2 is observed at
a redshift of 1.5, i.e. its comoving radial distance is about 4.4 Gpc.
Assume that the quasar in the previous question is observed to have a
companion galaxy which is 5 arcseconds apart. What is the projected...
I'm designing a pivot lift system to lift my movie projector up when not in use. I've designed the parts and begun 3D printing, but am concerned that the 3D print won't be strong enough, so I may bite the bullet and have the parts CNC machined. If I do that I want to be certain that I have the...
I came across the following problem somewhere on the web. The original site is long gone.
The problem has me stumped. May be sopmeone can provide some insight.
(The problem seems too simple to post in the "Linear/Abstract Algebra" forum.)
The Cost of Beer
It was nearing Easter, and a group...
I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...
When you write out the equations of motion for a system of two isolated charges, you can add both of the equations and get the increase in the particles linear momentum on one side. On the other side, you get the sum of all the forces between the particles. I understand that this sum of forces...
My angular acceleration is wrong but all I had done was torque which was 110 NM / I = 930 kg-m^2 and calculated 0.118 rad/s^2. But because this is wrong I am stuck and I have no idea how to find angular velocity to plug into the equation to find linear acceleration.
Hi Guys, please please assist me!
I need to pick for a project at my work linear carriages which should work at this configuration :
As you can see, there is a piston which moves the plate on 4 bearings from the side (yes, I know, not the best configuration, but that's what we have given the...
Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and
\begin{align}
\left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
Hi all, I have a problem on linear estimation that I would like help on. This is related to Wiener filtering.
Problem:
I attempted part (a), but not too sure on the answer. As for unconstrained case in part (b), I don't know how to find the autocorrelation function, I applied the definition...
Hello,
Models can be linear and nonlinear and I just learned that a "linear model" is not just a model where the dependent variable ##Y## is connected to independent variables ##X## raised to the 1st power... The model is called "linear" because the coefficients/parameters are not raised to...
[This is a continuation of OP's thread here: https://www.physicsforums.com/threads/satellite-mechanics-linear-and-rotational-momentum.1046963/ ]
satellite mechanics: linear and rotational momentum
I'm trying to better understand classical mechanics, and came up with a question:
Say we have a...
satellite mechanics: linear and rotational momentum
I'm trying to better understand classical mechanics, and came up with a question:
Say we have a squared satellite weighting 100kg, 1 meter on each side. it has a thruster on it's side, shown in picture
thruster quickly ejects 100g of propellant...
I need a row of switches, about 30 to be exact and the idea I need is to have a "moving OFF position" much the same way a "LED chaser" works where you have a moving ON position.
Only unlike in a LED chaser where one diode comes ON at any given time and the positions changes continually to...
I shall not begin with expressing my annoyance at the perfect equality between the number of people studying ODE and the numbers of ways of solving the Second Order Non-homogeneous Linear Ordinary Differential Equation (I'm a little doubtful about the correct syntactical position of 'linear')...
Okay so far I have come up with the following:
The objective function is:
30LADA + 40LAHO + 50SADA + 40SAHO + 110DANY + 125DACH + 105HONY + 115HOCH (to read this please see the table above as I have just used the first two letters of each except NY which I have represented by NY). In these I...
So far I have figured out the following:
maximize 0.02x1 + 0.01x2 +0.06x3 + 0.07x4
subject to:
xi <= 200,000
xi >= 24,000
x3 <= x2 +x4
x1 >= x3
But when I try to solve this in solver, I get an error which means my contraints must be wrong. Any suggestions on what constaints to put?
Suppose, for a certain a repo, I create a branch protection rule for the main branch with the following:
Require pull request before merging. This means that each feature or bugfix must be on separate branches, which will be later merged with main.
Maintain linear history of the main branch...
Hello! I have some data points ##(x,y)##, with some uncertainties on y that are statistical ##dy_1## and some systematic ##dy_2##. I want to fit this data using a linear function. How exactly should I deal with the different types of uncertainties? Can I just add them in quadrature and perform...
Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2?
____________________________________________________________
An ordered basis for C2x2 is:
I don't...
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.
Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)?
I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...
I am trying to convert the attached picture into dirac notation.
I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ>
The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS.
*Was going to type in LaTex but I...
The point (1, 5) is on the curve: y=ax^2+bx+c. This point gives the linear equation: 5 = a + b + c. A second point on the curve, (2, 10) gives the linear equation 10=4a+2b+c. A student called Erika thinks that the point (2, 19) is also on the curve.
5 = a + b + c.
10=4a+2b+c
19=4a+2b+c
the...
So this expression is apparently in Sz basis? How can you see that?
How would it look in Sy basis for example?
The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here?
Thanks