What is Orthogonal: Definition and 581 Discussions

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.

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  1. M

    Seeing Orthogonal views in 3-D?

    I'm having a hard time envisioning what the orthogonal view of this object in 3-D. It is mainly because of the continuous surface on the front view. Is that way in the back and the there are rectangular blocks sticking out? Thank you. I would have an easier time if a right view was included but...
  2. Y

    Orthogonality and orthogonal set.

    I am brushing up this topic. I want to verify both orthogonality between two functions and an orthogonal set ALWAYS have to be with respect to the specified interval...[a,b]. That is, a set of {1, ##\cos n\theta##, ##\sin m\theta##} is an orthogonal set IF AND ONLY IF ##\theta## on...
  3. B

    Orthogonal Coordinate Systems

    Hey guys, I'd really love it if you could post little essays explaining your intuition on how to derive the x, y & z coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, bipolar coordinates if you had to re-derive the coordinate system on a desert...
  4. S

    How to count all the orthogonal transformations?

    Homework Statement Find an orthogonal transformation ##\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}## that map plane ##x+y+z=0## into ##x-y-2z=0## and vector ##v_{1}=(1,-1,0)## into ##(1,1,0)##. Count all of them! Homework Equations ##A_{S}=PA_{0}B^{-1}##The Attempt at a Solution So basis...
  5. P

    Ricci tensor of the orthogonal space

    While reading this article I got stuck with Eq.(54). I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the result from the Gauss embedding equation and the Ricci identities for the 4-velocity, u^a. Is the...
  6. J

    Orthogonal vector equation (Ax=b)

    Hi I'm having problems understanding vector representation in the form Ax=B could someone please point me in the right direction A vector equation for a given straight line is r = (i + 3j) + t(-i-j). i) Show that the point (1,2) does not lie on this line. ii) Construct a vector...
  7. L

    Orthogonal Complements of Linear Transformations with Matrix A

    Homework Statement Let L: ℝn→ℝm be a linear transformation with matrix A ( with respect to the standard basis). Show that ker(L) is the orthogonal complement of the row space of A. Homework Equations The Attempt at a Solution The ker(L) is the subset of all vectors of V that map...
  8. L

    Proving Orthogonal Bases Homework Statement

    Homework Statement Let B be an ordered orthonormal basis for a k-dimensional subspace V of ℝn. Prove that for all v1,v2 ∈ V, v1·v2 = [v1]B · [v2]B, where the first dot product takes place in ℝn and the second takes place in ℝk. Homework Equations The Attempt at a Solution Let B...
  9. E

    Proving complex exponentials are orthogonal.

    I was wondering how you prove that ∫(e^iax)(e^ibx)dx from minus infinity to infinity is zero. When I try to evaluate this in the usual way, the result is undefined. Thanks in advance for your help!
  10. Petrus

    MHB The Orthogonality of Vectors and Matrices

    Hello MHB, I wounder if I did understand correct, If we got 3 vector and they all are orthogonala, v_1=(x_1,y_1,z_1),v_2=(x_2,y_2,z_2),v_3=(x_3,y_3,z_3) does that also mean that the matrix orthogonal so the invrese for the matrix is transport? Regards, |\pi\rangle
  11. Petrus

    MHB Orthogonal Rotation: Get Info & Tips

    Hello MHB, Do anyone know any good page that give you good describe when you rotate with orthogonal. I mean when you rotate base or vector in a orthogonal base ( hope this make sense) cause I did not understand from my book :( Regards, |\pi\rangle
  12. Petrus

    MHB Calculating Orthogonal Projection: Proving AD and Formula Progress | \pi\rangle

    Hello MHB, I have hard to prove that AD, I did put on pic the formula for AD and my progress is at bottom. Regards, |\pi\rangle
  13. M

    Stern-Gerlach experiment with orthogonal spin detector

    Suppose we have a Stern-Gerlach apparatus through which we send spin-1/2 particles and subsequently measure their position. If a passing particle "collapses" to the spin up state about X, it moves "up" and is registered by detector U, otherwise it "collapses" to the spin down state about X and...
  14. O

    Finding orthogonal of two vectors

    Homework Statement A vector in lR 3 (basis) has vector space V with the standard inner product. I need to find a vector in V which is perpendicular to both vectors v_1 = (1,2,1)^T and v_1 = (2,1,0)^T Homework Equations There is no real important equations other than just using...
  15. MarkFL

    MHB Nikachu's question at Yahoo Answers regarding orthogonal trajectories

    Here is the question: Here is a link to the question: Find the family of functions (circles) ? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response. edit: I see the question has been deleted at Yahoo!. (Dull)
  16. A

    Orthogonal polynomials are perpendicular?

    Orthogonal polynomials are perpendicular?? hi.. So as the title suggests, i have a query regarding orthogonal polynomials. What is the problem in defining orthogonality of polynomials as the tangent at a particular x of two polynomials are perpendicular to each other, for each x? This...
  17. K

    Wigner function of two orthogonal states: quantum harmonic oscillator

    The Wigner function, W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty} \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; , of the quantum harmonic oscillator eigenstates is given by, W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; , where \epsilon =...
  18. Z

    Vectors in R^4 orthogonal to two vectors

    Homework Statement Find all vectors in $\mathbb R^4$ that are orthogonal to the two vectors $u_1=(1,2,1,3)$ and $u_2=(2,5,1,4)$. Homework Equations Gauss-elimination. Maybe cross-product or Gram Schmidt. The Attempt at a Solution a) Denote a vector $u_3=(v_1,v_2,v_3,v_4)$ My...
  19. M

    Linear algebra orthogonal compliment

    Homework Statement Hello, I took my quiz today, and had to find a basis for an orthogonal compliment, would it be incorrect to not factor out the alphas and betas? Homework Equations The Attempt at a Solution
  20. cocopops12

    Do all complex functions have orthogonal real and imaginary parts?

    z = h(x) + ig(x) True or False: By the definition of the complex plane, h(x) and ig(x) will always be orthogonal. If this was true, wouldn't that mean that we can find a 'very general' Fourier series representation of any function f(x) as an infinite series of An*h(x) + infinite series of...
  21. M

    Linear algebra question: Orthogonal subspaces

    Homework Statement For each of the following matrices, determine a basis for each of the subspaces N(A) A=[3 4] [ 6 8]Homework Equations The Attempt at a SolutionSo reducing it I got [1 4/3] [0 0] I know x2 is a free variable I set x2 = to β and found...
  22. V

    Solving Orthogonal Families Homework Problem

    Homework Statement Homework Equations The product of the slopes has to be equal to -1. The Attempt at a Solution Well, as the function, call it u, is equal to a constant, the derivative of u with respect to x is the partial derivative with respect to x + (the partial derivative of u with...
  23. C

    Coupled mass problem with orthogonal oscillations

    Homework Statement Consider a light elastic string of unstretched length ##4a_o##, stretched horizontally on a smooth surface between two fixed points a distance ##4a## apart. (##a > a_o)##. Three particles of mass m are attached so as to divide the string into four equal sections. Number...
  24. M

    Determining the general form of an orthogonal vector

    Homework Statement Determine a vector that is orthogonal to both (1,2,-1) and (3,1,0) Homework Equations As above. The Attempt at a Solution The solution, from the back of the book, is "any vector of the form (a, -3a, -5a), but I'm not sure how they got there. I get the...
  25. K

    Finding orthogonal vector

    Homework Statement Find a unit vector that is orthogonal to both u=(1,1,0) and v=(-1,0,1) Any help appreciated thanks!
  26. F

    Finding vectors orthogonal to the span of a matrix

    Homework Statement My linear algebra is rusty. So to go from a reduced QR factorization to a complete QR factorization (ie the factorization of an over determined matrix) one has to add m-n additional orthogonal vectors to Q. I am unsure on how to find these. If it is extending a 3x2 to a 3x3...
  27. M

    Question in orthogonal trajectories

    Hi answer b I forget first step 2aydy/dx = 3x^2 please can check my answer
  28. M

    Find the orthogonal trajectories

    Hi all
  29. A

    Find all unit vectors orthogonal to the line

    Homework Statement Find all the unit vectors orthogonal on the line L. Homework Equations L passes through the vectors u = [9; 7] and v = [1; -5] The Attempt at a Solution I found the slope of L from the two vectors: 3/2. I know that to be orthogonal, the vector will have a...
  30. H

    When the Curl of a Vector Field is Orthogonal

    Simple question. It came out of lecture, so it's not homework or anything. My professor said that the curl of a vector field is always perpendicular to itself. The example he gave is that the magnetic vector potential A is always perpendicular to the direction of the magnetic field B. (I haven't...
  31. S

    Prove the intersection of two orthogonal subspaces is {0}

    Homework Statement Let A and B be two orthogonal subspaces of an inner product space V. Prove that A\cap B= \{ 0\}. Homework Equations The Attempt at a Solution I broke down my proof into two cases: Let a\in A, b\in B. Case 1: Suppose a=b. Then \left\langle a,b \right\rangle =...
  32. V

    Inner Product Space of two orthogonal Vectors is 0 , Is this defined as it is ?

    This may be a very silly question, but still apologies, I read in Sheldon Axler, that the inner product of two orthogonal vectors is DEFINED to be 0. Let u,v belong to C^n. I am unable to find a direction of proof which proves that for an nth dimension vector space, if u perp. to v, then <u,v>...
  33. shounakbhatta

    Orthogonal set - Geometric interpretation

    Orthogonal set -- Geometric interpretation Hello, If we have two vectors u,v then in an inner product space, they are said to be orthogonal if <u,v>=0. Well, orthogonal means perpendicular in Euclidean space, i.e. 90 degrees. How <u,v> becomes zero. Secondly, if I have three vectors...
  34. L

    Find orthogonal vector to current vector in 3D

    Hi, In 2D I know a simple answer: vector (a,b) is orthogonal to vector (-b,a) Is there anyway similar to that to find an orthogonal vector in 3D?
  35. E

    Approximation in orthogonal functions with pulse shaping

    Hello! I have tried for a whole afternoon to solve this problem but I didn't succeed. Let \cos(2 \pi (f_0 + i/T_N) t + \phi_i) and \cos(2 \pi (f_0 + j/T_N) t + \phi_j) be two quasi-orthogonal functions: \int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt...
  36. C

    Proving Orthogonality Between Row and Column Vectors of Invertible Matrices

    Let A be an n × n invertible matrix. Show that if i ≠ j, then row vector i of A and column vector j of A-1 are orthogonal. I'm lost in regards to where to lost. I want to show that a vector from row vector i from A is orthogonal to a column vector j from A. Orthogonal means the dot...
  37. C

    How do you check if 2 vectors are orthogonal?

    How do you check if 2 vectors are orthogonal? I know that if 2 vectors are orthogonal, then there dot product is 0. But I don't think that necessarily means if their dot product is 0, the 2 vectors are orthogonal. Like what if you had 2 zero vectors, their dot produt would be 0, but they're not...
  38. J

    Standard Matrix for an orthogonal projection transformation

    Let T:R^2 -> R^2 be the linear transformation that projects an R^2 vector (x,y) orthogonally onto (-2,4). Find the standard matrix for T. I understand how to find a standard transformation matrix, I just don't really know what it's asking for. Is the transformation just (x-2, y+4)? Any...
  39. djh101

    Orthogonal Projections: Minimize a^2 + b^2 + c^2

    Homework Statement There are three exams in your linear algebra class and you theorize that your score in each exam will be numerically equal to the number of hours you study. The three exams count 20%, 30%, and 50% and your goal is to score 76% in the course. How many hours, a, b, and c...
  40. N

    Orthogonal projection question

    Homework Statement Hello, H is a Hilbert space. K is a nonempty, convex, closed subset of H. Prove that the orthogonal projection Pk: H → H, is non-expansive: ll Pk(x) - Pk(y) ll ≤ ll x - y ll The Attempt at a Solution So the length between the Pk's, which is in K (convex) is less than...
  41. T

    Writing orthogonal vectors as linear combinations

    Hello, Quick question, not really homework but more of a general inquiry. Take three vectors: a,b and c such that a and c are orthogonal. Is it possible to write c as a linear combination of a and b such that: c = ma + nb where m,n are scalars. I was thinking not at first glance but...
  42. G

    Line maximizing orthogonal projections

    Say I have a set of points in 2D space. How would I find a line that maximizes the sum orthogonal projection of the points onto the line. The line does not have to go through the origin.
  43. C

    Curvature of an orthogonal projection

    Homework Statement Let \vec{X(t)}: I \rightarrow ℝ3 be a parametrized curve, and let I \ni t be a fixed point where k(t) \neq 0. Define π: ℝ3 \rightarrow ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to \vec{X(t)} at t. Define γ=π\circ\vec{X(t)} as the orthogonal projection...
  44. ElijahRockers

    Orthogonal Projection of vector Y onto subspace S

    Homework Statement Let S be the linear span of the orthogonal set: {[3 2 2 2 2]T,[2 3 -2 -2 -2]T,[2 -2 3 -2 -2]T} Calculate the orthogonal projection of Y = [1 2 -1 3 1]T onto S. The Attempt at a Solution Not sure how to go about this... Do i find a vector that is orthogonal...
  45. O

    Transforming a matrix to orthogonal one

    Suppose a matrix X of size n x p is given, n>p, with p linearly independent columns. Can it be guaranteed that there exists a matrix A of size p x p that converts columns of X to orthonormal columns. In other words, is there an A, such that Y=XA, and Y^TY=I, where I is an p x p identity matrix.
  46. T

    Interior and boundary of set of orthogonal vectors

    Let "a" be a non zero vector in R^n and define S = { x in R^n s.t. "a" · "x" = 0}. Determine S^int , bkundary of S, and closure of S. Prove your answer is correct Attempt: Ok I am more sk having trouble proving that the respective points belong to its condition. Such as thr...
  47. estro

    Linear Algebra - orthogonal vector fields

    I want to prove that: Ker(T*)=[Im(T)]^\bot Everything is in finite dimensions. What I'm trying: Let v be some vector in ImT, so there is v' so that Tv'=v. Let u be some vector in KerT*, so T*u=0. So now: <u,v>=<u,Tv'>=<T*u,v'>=0 so every vector in ImT is perpendicular to every vector...
  48. D

    Finding ODE for Family of Orthogonal Curves to Circle F

    Homework Statement Consider the family F of circles in the xy-plane (x-c)2+y2=c2 that are tangent to the y-axis at the origin. What is a differential equation that is satisfied by the family of curves orthogonal to F? Homework Equations ∇f(x,y)=<fx,fy> The Attempt at a SolutionMy general...
  49. J

    Truly Bizarre - The unit tangent and unit normal vectors aren't orthogonal

    OK, this looks like a differential geometry problem, which it is, but at the end of the day I am trying to figure out why the unit normal and unit tangent vectors to a curve aren't orthogonal, so even if you don't know about DG, please respond. Obviously the two choices for E_1 and E_2...
  50. P

    Give a 2x2 matrix with Det(A)=+/-1 that is not orthogonal.

    I attached the problem. I only need help with part b), I provided part a) just to remind you guys what a orthogonal matrix was. The only 2x2 matrix I can think of with a determinant of + or - 1 is something like √1 0 0 1 The determinant of this would be √1 = + or - 1 The...
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