What is Linear: Definition and 1000 Discussions

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".

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  1. J

    Linear Transformation from R3 to R3

    "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.
  2. murshid_islam

    What would be a good book for learning Linear Algebra by myself?

    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...
  3. guyvsdcsniper

    Linear Algebra in Dirac Notation

    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...
  4. E

    B Attempt to solve this system of three linear equations

    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...
  5. Graham87

    Intro to quantum mechanics - Spin and linear algebra

    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
  6. M

    Obtain the eight incongruent solutions of the linear congruence

    Consider the linear congruence ## 3x+4y\equiv 5\pmod {8} ##. Then ## 3x\equiv 5-4y\pmod {8} ##. Note that ## gcd(3, 8)=1 ## and ## 1\mid (5-4y) ##. Since ## 3^{-1}\equiv 3\pmod {8} ##, it follows that ## x\equiv 15-12y\pmod {8}\equiv 7+4y\pmod {8} ##. Thus ## {(x, y)=(7+4y, y)\pmod {8}\mid 0\leq...
  7. TGV320

    Courses Linear Algebra Self-Study: Textbook & Quantum Mechanics

    Hello, I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra. Some textbooks in my country are quite bad and only teach...
  8. P

    Absorption coefficient and Linear Optical Susceptibility

    ##\alpha## is considered to be the absorption coefficient for a beam of light of maximum intensity ##I_0##. It's related to the complex part of the refractive index as we have shown above. Now, I have a doubt. Should I solve for ##k## from the quadratic equation in terms of the linear optical...
  9. chwala

    Solve the following linear system

    Find the problem below with the solution indicated; The text approach is much clear to me... My way of tackling the problem is as follows; using echelon form (row reduction method) \begin{bmatrix} 1 & 1 & 2 & -5 &3\\ 2 & 5 & -1 & -9&-3 \\ 2 & 1 & -1 & 3&-11\\ 1& -3 & 2 & 7&-5...
  10. T

    What are the two different senses of since and how are they used?

    Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus? Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
  11. M

    A Piecewise linear interpolation with uncertainties

    Hello! I have a function ##y = f(x_1,x_2)##, and I would like to do a piecewise linear interpolation. However, both the dependent (##y##) and independent variables (##x_1, x_2##) have uncertainties associated to them (the uncertainty is the same for a given variable i.e. all ##x_1## measurements...
  12. C

    I Summing linear momentum in circular rotation

    Hello everyone! I was wondering why can't we take a rotating body and see the linear movement that each particle moves to find the 'total linear momentum,' I imagine this quantity would be conserved, and furthermore couldn't you write the total linear momentum as a function of angular velocity...
  13. Ahmed1029

    Griffith's problem 4.37 -- A point dipole at the center of a sphere of linear dielectric material

    I tried to solve it for some time and then looked at the solution manual, which got me completely lost. Those are the first lines of the solution : I'm not so sure how equation 4.39: makes him conclude that the same relation holds for dipole moments. My second concern is that I'm not sure how...
  14. A

    Effects of brownouts and power spikes on linear power supplies

    What are the effects of brownouts and power spikes on linear power supplies in particular; e.g., a linear power supply that provides 24 V to an inductive angle encoder? Thanks for any information you could provide.
  15. B

    I Uncertainties from linear interpolation

    Hello! I have a function of several variables (for this questions I assume it is only 2 variables), ##y = f(x_1,x_2)##. I want to learn this function using simulated data (i.e. generated triplets ##(x_1,x_2,y)##) and then use that function to get ##y## from measured ##(x_1,x_2)##. There is no...
  16. MidgetDwarf

    Analysis Book needed for Applied Linear Analysis Course

    The course description is as follows: Description Function spaces, convergence, inner product, bounded linear operators, integral operators and integral equations, adjoint operators, expansion in eigenfunctions, resolvent, kernel The instructor is on sabbatical, until the start of the class...
  17. Salmone

    I Linear independence of three vectors

    If I've got three vectors ##\vec{a}##, ##\vec{b}## and ##\vec{c}## and ##\vec{a}##, ##\vec{b}## are linearly independent and ##\vec{c}## is linearly independent from ##\vec{a}##, is ##\vec{c}## also linearly independent from ##\vec{b}##?
  18. R

    I How to account for linear momentum in a collision?

    Suppose a bar is fixed to an axle at one one so that it can pivot. The bar is initially motionless, but is set rotating about it's axle when impacted by a ball. (The ball does not strike the bar at it's pivot point.) Suppose the collision is such that the bar is set rotating and the ball is...
  19. Salmone

    I Linear combination of states with Pauli's principle

    If I have two identical particles of ##1/2## spin, for Pauli's exclusion principle all physical states must be antysimmetrical under the exchange of the two particles, so ##\hat{\Pi}|\alpha\rangle=-|\alpha\rangle##. Now, let's say for example this state ##\alpha## is an Hamiltonian eigenfunction...
  20. J

    Max linear speed of propeller tip in water

    Homework Statement:: project not homework Relevant Equations:: vectors Hi,I am trying to determine the maximum linear velocity of a propeller tip when subject to flowing water with velocity ##\vec V##. For simplicity, I will assume that rotational inertial is negligible. The drawing below...
  21. shivajikobardan

    Comp Sci Can't we use linear regression for classification/prediction?

    they say that linear regression is used to predict numerical/continuous values whereas logistic regression is used to predict categorical value. but i think we can predict yes/no from linear regression as well Just say that for x>some value, y=0 otherwise, y=1. What am I missing? What is its...
  22. isaacdl

    A Proving $g(u,v)≠0$ with Linear Independence

    I'm trying to prove that there exist always a vector w whose contraction with a lightlike vector u (g(u,u)=0) it's always different from zero: $g(u,v)≠0$I know how to do this with coordinates, but in a free cordinate scheme I'm totally lost. Any help? PD: Both vectors are linearly independent.
  23. M

    A Linear combination of data with uncertainty

    Hello! I have 2 measured data points (they are measurements of different observable, not 2 measurement of the same observable), with quite different errors, say ##x_1 = 100 \pm 1## and ##x_2 = 94 \pm 10##. I want to compute the value (and associated uncertainty) of a linear combination of them...
  24. M

    Engineering First order non linear to state space equations

    How to represent this system in state space form? where: $$ x' = Ax + Bu \text{ and }y = Cx + Du$$ I am trying to create a state space model based on these equations on simulink, need to find A, B, C and D but like I mentioned, i cannot find the solution when the differentials are not of...
  25. chwala

    Solve the problem involving linear programming

    Find question and solution here; The initial steps were a bit confusing to me...i decided to use hours instead of minutes ...only then did it become more clear to me. See my graph, Ok i follow that the function would be optimised at ##x=45## and ##y=6.25## ...now to my question...we...
  26. H

    Finding the angle at the apex of a rhomb with incoming linear wave

    Hi, Since I'm dealing with a rhombus, the angle at the bottom(A) and top(A) are the same. Thus, I only have to find the angle at the bottom since the incoming beam is already perpendicular to the side of the rhombus. Since the incoming beam is perpendicular to the side ##\theta_I = \theta_T =...
  27. L

    A Propagation modes and linear systems

    In the book "Fundamentals of photonics", the authors defined waveguide modes using the notion of linear systems, where they said: "Every linear system is characterized by special inputs that are invariant to the system, i.e., inputs that are not altered (except for a multiplicative constant)...
  28. H

    I Understanding the Expression for a Linear EM Wave Transmission?

    Hi, I have an expression in my textbook that I don't really understand. I have 2 questions regarding this expression for a linear EM wave## \tilde{\vec{E_{0i}}} = (E_{0x} \hat{x} \pm E_{0y} \hat{y}) e^{i(kz- \omega t)}## ## \tilde{\vec{E_{0t}}} = (\sum_j E_{oij} e_{pj}) \hat{e_p} ## ##...
  29. ChichoRD

    Creating an Enemy with 2D Linear Movement: Calculating Projectile Angle

    Hi, I am a game developer and recently found myself in a situation where I wanted to create an enemy that shoots at you given a set of variables. The game takes place in a top down view so there is no need for gravity acceleration, just 2d linear constant movement. The question to answer is...
  30. G

    Force on a particle of a linear charge distribution

    Hello! I am trying to solve this exercise of the electric field, but it comes out changed sign and I don't know why. Statement: On a straight line of length ##L=60\, \textrm{cm}## a charge ##Q=3,0\, \mu \textrm{C}## is uniformly distributed. Calculate the force this linear distribution makes...
  31. S

    Quick-adjusting linear actuator?

    I am making a project which requires a linear actuator which can be adjusted manually, in a similar manner to those Vices which have the ratchet lever, allowing you to manually push the vice in or pull it out rather than winding it in and out. I'd like the same principle but on a linear actuator...
  32. F

    Relating Linear and Angular Kinematics

    a) We use the definition of linear speed in terms of angular speed: v = r*omega omega_f = v/r = (1.25 m/s)/(0.025 m) = 50 rad/s omega_0 = v/r = (1.25 m/s)/(0.025 m) = 21.55 rad/s b) We use the definition of linear speed: v = d/t d = vt = (1.25m/s)(74 min)(60 s/1 min) = 5.55 km c) We use the...
  33. G

    B A moving magnet in a linear electric field

    If a electrically charged mass travels thru a magnetic(m) field, it will accelerate at right angles to its velocity and the m-field. Under some conditions like this the charged mass will travel in a circular loop due to this magnetic force acceleration. This info is all over the internet. e.g...
  34. H

    MATLAB Modelling a Heat Pump with Linear Compressor using MATLAB Simulink

    Hello, I am a final year mechanical engineering student designing a heat pump with a linear compressor for an electric vehicle and have decided to model this using Simulink, I would love some help on the governing equations/formulas needed for each component of the heat pump as well as guidance...
  35. A

    Engineering Signals & Systems with Linear Algebra

    Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks! My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
  36. S

    Codomain and Range of Linear Transformation

    Standard matrix for T is: $$P=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & -1 \end{bmatrix}$$ (i) Since matrix P is already in reduced row echelon form and each row has a pivot point, ##T## is onto mapping of ##\mathbb R^3 \rightarrow \mathbb R^2## (ii) Since there is free variable in matrix P, T is...
  37. L

    Linear systems: Tmax = Umax is not making sense

    We have a slide in class that states if no friction or damping force, then the system is conservative. Then it shows: delta(T+U)=0 or T+U=constant. It then goes on to say that max kinetic energy is equal to max potential energy which is false. no way can you have KEmax=Pemax... I double...
  38. P

    I Proof that two linear forms kernels are equal

    Attempt of a solution. By the Rank–nullity theorem, $$ \dim V=\dim Im_{F}+\dim\ker\left(F\right) \Rightarrow n=1+\dim\ker\left(F\right) \Rightarrow \dim\ker\left(F\right)=n-1. $$ Similarly, it follows that $$\dim\ker\left(G\right)=n-1.$$ This first part, for obvious reasons, is very clear. The...
  39. H

    Prove that the linear space is infinite dimensional

    A space is infinite dimensional when its basis is infinite. But how can I ensure that the basis of the space of all sequences whose limit is zero is infinite? (After solving that, I would like to have a hint on this very similar problem which says: let V be a Linear space of all continuous...
  40. bobtedbob

    I Linear Bezier curve projection

    I'm looking at the following web page which looks at rendering bezier curves. GPU Gems 3 - Chapter 25 Paper on same topic The mathematics is quite interesting, I was interested to know what the F matrix would look like for for a linear bezier equation. The maths for the quadratic case is in...
  41. C

    One set v is a linear combination of u. Prove u is linearly dependent

    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...
  42. H

    Prove that T is a linear transformation

    We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, : Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?) OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...
  43. H

    The correct way to write the range of a linear transformation

    We have a transformation ##T : V_2 \to V_2## such that: $$ T (x,y)= (x,x) $$ Prove that the transformation is linear and find its range. We can prove that the transformation is Linear quite easily. But the range ##T(V_2)## is the the line ##y=x## in a two dimensional (geometrically) space...
  44. Vectronix

    Linear Algebra I need a book on linear algebra....

    Is Advanced Linear and Matrix Algebra by Nathaniel Johnston a good book on linear algebra? Will it teach me all I need to know? Is there any calculus in it despite the name? I never took a course on linear algebra so I'm looking for something that teaches everything and includes calculus with...
  45. O

    I Linear Accelerator Length Contraction

    I am trying to understand the effect of relativistic length contraction on the electron bunches in a linear accelerator. Figure B is for nonrelativistic speeds, successive cylinder lengths are progressively longer. However, wikipedia says "At speeds near the speed of light, the incremental...
  46. U

    I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

    Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
  47. U

    I Limit of limits of linear combinations of indicator functions

    I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
  48. terrytosh

    Should I study Analysis before Linear Algebra?

    Or is reading a proofs book enough
  49. chwala

    Classify the given second-order linear PDE

    Now i learned how to use discriminant i.e ##b^2-4ac## and in using this we have; ##b^2-4ac##=##0-(4×3×2)##=##-24<0,## therefore elliptic. The textbook has a slight different approach, which i am not familiar with as i was trained to use the discriminant at my undergraduate studies... see...
  50. S

    I Linear Algebra 1 problem, Vector Geometry: Lines

    Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...
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