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. N

    Proof that v'(t) is orthogonal to v(t)

    Homework Statement Prove that if v(t) is any vector that depends on time, then v'(t) is orthogonal to v(t). Hint given: Consider the derivative of v^2. The Attempt at a Solution V'^2 = d/dt (v * v) = v d/dt + v d/dt = d/dt (v+v) = 2v d/dt ??
  2. D

    Showing orthogonal polynomials are unique

    Homework Statement We are given that a set of polynomials on [-1,1] have the following properties and have to show they are unique by induction. I have a way to show they are unique, but is not what he is looking for. I honestly have never seen it presented this way. P_n(x) = Ʃa_in*x^i All...
  3. S

    A question on the orthogonal polynomial

    Dear All Friends, I am currently working on a project which needs some orthogonality integration formulae of Laguerre polynomials. I referred worlfram's math function site http://functions.wolfram.com/Polynomials/LaguerreL3/21/02/01/ and get three seemingly useful ones...
  4. G

    Orthogonal Trajectory. Calc IV.

    Homework Statement Find orthogonal trajectories for y=clnx, y=cex, y=sin(x)+cx Homework Equations Simple integration for the most part. The Attempt at a Solution I'm fairly confident on the first two, its the third that's giving me trouble. First, y=cln(x) c=\frac{y}{ln(x)}...
  5. T

    Why is the special orthogonal group considered the rotation group?

    I understand that the special orthogonal group consists of matrices x such that x\cdot x=I and detx=1 where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule x\cdot x=I are matrices involved with rotations because they preserve the dot...
  6. G

    Find normalised linear combinations that are orthogonal

    Homework Statement I'm a little weary of posting this in this forum. If I post it in the math section it will be answered in about 30 min whereas here it might take about 5 hours, but we'll see. Homework Equations The Attempt at a Solution Number one, I'm not exactly sure how they get from...
  7. F

    Finding the Eigenstuff of a Orthogonal Projection onto a plane

    Homework Statement Let S be the subspace of R3 defined by x1 - x2 + x3 = 0. If L: R3 -> R3 is an orthogonal projection onto S, what are the eigenvalues and eigenspaces of L? Homework Equations The Attempt at a Solution First off, I hadn't seen the term eigenspace before. From...
  8. T

    Is this a complete test to show that a matrix is orthogonal?

    I used to test orthogonality by using the definition MT = M-1, which means I always calculated the inverse of the matrices. However, isn't it true that if M is orthogonal, then MMT = I? If we multiply both side by M-1, we get MT = M-1. Can I use this to proof the orthogonality of a matrix...
  9. J

    Finding an Orthogonal Base for Vector Space H

    Homework Statement Greetings, I'm trying to solve these problems given the vectors u=(3,-2,1) and v=(2,-3,1) 1. Find an orthogonal base for the space H generated by {u,v} 2. Find the orthogonal projection of w=(3,0,1) on H Homework EquationsThe Attempt at a Solution Im not sure how...
  10. G

    Determine if the given vectors are orthogonal

    Homework Statement Homework Equations The Attempt at a Solution A set of vectors are orthogonal if any two are perpendicular. the cross product of w1 and w2 is -9 + 2 + 3 + 4 = 0 So the set of vectors is orthogonal. The book says that's false. Why?
  11. A

    Orthogonal properties of associated laguerre polynomial

    i need the derivation of orthogonal properties of associated laguerre polynomial (with intermediate steps). someone please tell me where can i get it (for easy understanding).
  12. Vorde

    Orthogonal Lines and their line element

    In one of the early chapters of Gravity by Hartle, he is developing the line element on a sphere in preparation for developing the concept of a spacetime interval. Whilst finishing up the proof Hartle sort of implicitly says that if two lines are orthogonal the line element connecting two points...
  13. M

    Show orthogonal matrices are manifolds (Munkres Analysis on Manifolds)

    Homework Statement Let ##O(3)## denote the set of all orthogonal 3 by 3 matrices, considered as a subspace of ##\mathbb{R}^9##. (a) Define a ##C^\infty## ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6## such that ##O(3)## is the solution set of the equation ##f(x) = 0##. (b) Show that ##O(3)## is a...
  14. L

    Confirm the row vectors of A are orthogonal to the solution vectors?

    HI there. I'm taking Linear Algebra classes right now and this question has been bugging me. Homework Statement Find a general solution to the system, state the dimension of the solution space, and confirm the row vectors of A are orthogonal to the solution vectors. The given system is: (x1)...
  15. S

    A conjugacy class under O(n), orthogonal projection

    This is not really a homework question per se but I wasn't sure where else to put it: In a script I'm reading the following set is defined: P(n)_k := \{p \in S(n) | p^2 = p, \text{trace } p = k\} (i.e. the set of all real orthogonal projection matrices with trace k). Now the following...
  16. H

    Orthogonal Basis for a subspace

    Homework Statement Let W = \begin{cases} \begin{pmatrix}x\\y\\z\\w\end{pmatrix} \in R^4 | w + 2x + 2y + 4z = 0 \end{cases} A)Find basis for W. B)Find basis for W^{\perp} C)Use parts (A) and (B) to find an orthogonal basis for R^4 with respect to the Euclidean inner product. Homework...
  17. S

    Right Hand Side Orthogonal Drawing

    Homework Statement I need to draw an orthogonal RHS of an object I have attached as a pdf file with the message for an assignment. My problem is I do not know what the RHS of the drawing would look like. I know how to use autocad to do drawings but I'm stuck when I need to envisage the...
  18. facenian

    Question about Orthogonal Polynomials

    Hello, I'm studing the hydrogen atom and I found an unified presentation of orhtogonal polynomials in the book by Fuller and Byron. I would like to learn more about it but in the same spirit(for physicits not for mathematicians). Can someone give some references where to find more?
  19. J

    A LinAlg Proof Involving Orthogonal Complement

    Homework Statement Here is the problem and my complete answer. Am I OK? Thanks! http://www.d-series.org/forums/members/52170-albums1546-picture8143.jpg Homework Equations The Attempt at a Solution
  20. C

    Linear Algebra: orthogonal components of a vector

    Homework Statement Let V = Gen{ [0;5;1;2], [4;0;-2;1], [5,1,0,1]}. Define u11=v1. Indicate the coordinates of u2, the orthogonal component of v2 to V1=Gen{u1. Homework Equations I know V has to be a vector space. If there is a subspace W with an orthogonal basis B={v1,...,vk} then the...
  21. Z

    Proving Orthogonal Projection and Norm using Inner Products

    Homework Statement Let U be the orthogonal complement of a subspace W of a real inner product space V. Have already shown that T is a projection along a subspace W onto U, and that V is the direct sum of W and U. The questions now says: show ||T(v)|| = inf (w in W) || v - w ||...
  22. L

    Proving a matrix is orthogonal.

    Homework Statement Question 10a of the attached paper. Homework Equations The Attempt at a Solution If a matrix is orthogonal, its transpose is its inverse. The inverse U^{-1} is defined by ƩU^{-1}ij Vj = uj I don't know how to go about proving this. Thanks for any help!
  23. B

    Implications of orthogonal clocks in rockets

    Hi. If two light clocks are put on a rocket at rest and then accelerated to relativistic velocities with one of the light clocks parallel to the direction of motion and one perpendicular, will one clock continue to measure the rate of change of time in the rest plane while the other one...
  24. A

    How to find the orthogonal basis?

    Can somebody help me how to approach this problem.I am having trouble finding the orthogonal basis.
  25. L

    Confusion between orthogonal sum and orthogonal direct sum

    For 2 vector spaces an orthogonal direct sum is a cartesian product of the spaces (with some other stuff) (http://planetmath.org/encyclopedia/OrthogonalSum.html ), and this orthogonal direct sum uses the symbol, \oplus. However, there's an orthogonal decomposition theorem...
  26. S

    Linear Algebra Proof of Span and orthogonal vector space

    Let w=span(w1, w2, ...,wk) where wi are vectors in R^n. Let dot product be inner product for R^n here. Prove that if v*wi=0 for all i-1,2,...,k then v is an element of w^upside down T (w orthogonal).
  27. I

    Finding an orthogonal complement without an explicitly defined inner product

    Homework Statement P5 is an inner product space with an inner product. We applied the Gram Schmidt process to the basis {1,x,x^2,x^3,x^4} and obtained the following result. {f1,f2,f3,f4,x^4+2} What is the orthogonal complement of P3 in P5 with respect to this inner product?Homework Equations...
  28. L

    Orthogonal complement of gradient field?

    I am doing my research in probability. I have found some probability distribution of a random variable X on the n dimensional unit sphere. Let b be a smooth and lipschitz vector field mapping X to R^n. I have also found that for all continuous differentiable function f mapping X to R, the...
  29. A

    MHB Proving Orthogonal Polynomials: A Weighted Integral

    Let \{ \phi_0,\phi_1,...,\phi_n\} othogonal set of polynomials on [a,b] n>0, with a weight function w(x) prove that \int_{a}^b w(x)\phi_n Q_k (x) \; dx = 0 for any polynomail Q_k(x) of degree k<n ? My work : I think there is a problem in the question since if we take x^2,x^3 on the...
  30. R

    Is the Set {cos x, cos 2x, cos 3x, ...} Orthogonal Using Integral Products?

    How would you prove, using the integral product, that the set of {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set?
  31. G

    Linear Algebra - dimension of orthogonal complement

    I've attached a copy of the problem and my attempt at a solution. This seems like a relatively straightforward question to me, but my answer seems to be the exact opposite of what the answer key says. I reach the conclusion that the answer is C, but the answer is apparently D. I'm...
  32. H

    Can Eigenvectors of the Same Eigenvalue Be Orthogonal in a 2x2 Matrix?

    This seems a simple question but I can't find the solution by myself. Please help. Say we have a 2 by 2 matrix with different eigenvalues. Corresponding to each eigenvalue, there are a number of eigenvectors. Q1. Could the eigenvectors corresponding to the same eigenvalue have different...
  33. B

    Prove Finite Orthogonal Set is Linearly Independent

    Folks, I am looking at my notes. Wondering where the highlighted comes from. Prove that a finite orthogonal set is lineaarly independent let u=(x_1,x_2,x_n) bee an orthogonal set set of vectors in an ips. To show u is linearly independent suppose Ʃ ##\alpha_i x_i=0## for i=1 to n...
  34. T

    Linear Algebra: Orthogonal Projections

    Hey first time poster here. Homework Statement Find the orthogonal projection projWy u1 = [-1; 3; 1; 1], u2 = [3; 1; 1; -1], u3 = [-1; -1; 3; -1], y = [1; 0; 0; 1] where {u1, u2, u3} is an orthogonal basis. Homework Equations yhat = [dot(y,U1)/dot(U1,U1)]U1 + ...
  35. sergiokapone

    Quantum entanglement of spin along multiple orthogonal axes

    I already asked this question on physics.stackexchange.com, but did not get the desired response. I am interested in the opinion of your community. Picture an entangled pair of spin 1/2-spin particles with total spin 0. In the diagram, particle 1 of the pair is moving to the left (-y), and...
  36. F

    Gram-Schmidt Method for orthogonal basis

    I have S= {(1,1,0,1) (1,0,-1,0) (1,1,0,2)} its one of the subset and second it T= {(x,y,z,2x-y+3z)} If you were to use Gram-Schmidt method to find the orthogoan basis for T who would you processed? I really don't understand this concept. I know from T , the hyperplane is 2x-y+3z so the...
  37. T

    Orthogonal Basis and Inner Products

    Homework Statement The Attempt at a SolutionSince A is a vector in V and since the A_i form a basis, we can write A as a linear combination of the A_i. We write A = x_1 A_1 + ... + x_n A_n. Thus, we have, <x_1 A_1 + ... + x_n A_n,A_i> = 0 = x_1 <A_1,A_i> + ... + x_n <A_n,A_i>. Because...
  38. Math Amateur

    Orthogonal Transformations _ Benson and Grove on Finite Reflection Groups

    I am reading Grove and Benson's book on Finite Reflection Groups and am struggling with some of the basic linear algebra. Some terminology from Grove and Benson: V is a real Euclidean vector space A transformation of V is understood to be a linear transformation The group...
  39. Shackleford

    Orthogonal Projection in Inner Product Space with Dimension 2 and Basis {1,x}

    I found a final answer online, but my vector is slightly different. I haven't been able to catch my mistake. I'm supposed to find the orthogonal projection of the given vector on the given subspace W of the the inner product space V. P1 has dimension 2 and basis = {1,x}...
  40. C

    Find two unit vectors orthogonal to both given vectors

    Hello, Could someone please review my work and see if it is correct. Thanks :smile: Homework Statement Find two unit vectors orthogonal to both given vectors. i + j + k, 3i + k Homework Equations The Attempt at a Solution So I used cross product and got A x B= i+2j-3k...
  41. O

    Unit vector orthogonal to plane

    Homework Statement Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-4, 5, 4), Q = (-1, 8, 7), and R = (-1, 8, 8). Homework Equations u = PQ = Q - P v = PR = R - P ans = uXv = PQ X PR The Attempt at a Solution so I did...
  42. G

    Escape of light perfectly orthogonal to black hole

    I have a fairly decent understanding of black holes, but have always had one curiosity that I haven't found a distinct answer to: If light, through whatever reaction, is emitted inside the event horizon of a black hole such that it is directed in a path exactly orthogonal to the black hole...
  43. U

    Orthogonal Complements of complex and continuous function subspaces

    Homework Statement I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces: 1) span{(1,0,i,1),(0,1,1,-i)}, 2) All constant functions in V over the interval [a,b] Homework Equations I know that for a subspace W of an...
  44. F

    Why the inner product of two orthogonal vectors is zero

    Why is the inner product of two orthogonal vectors always zero? For example, in the real vector space R^n, the inner product is defined as ||a|| * ||b|| * cos(theta), and if they are orthogonal, cos(theta) is zero. I can understand that, but how does this extend to any euclidean space?
  45. Dembadon

    Linear Algebra I: Orthogonal Matrix Condition

    I would like to check my reasoning for this problem to make sure I understand what an orthogonal matrix is. Homework Statement Determine if the matrix is orthogonal. If orthogonal, find the inverse. \begin{pmatrix} -1 & 2 & 2\\ 2 & -1 & 2\\ 2 & 2 & -1 \end{pmatrix} Homework...
  46. T

    Finding Equation of an Orthogonal Line

    Homework Statement Let L1 be the line (0,4,5) + <1,2,-1>t and L2 be the line (-10,9,17) + <-11,3,1>t a) Find the line L passing through and orthogonal to L1 and L2 b) What is the distance between L1 and L2 The Attempt at a Solution I only know how to do part of part a). I can only find the...
  47. 3

    Eigenvalue for Orthogonal Matrix

    Homework Statement Let Q be an orthogonal matrix with an eigenvalue λ_{1} = 1 and let x be an eigenvector belonging to λ_{1}. Show that x is also an eigenvector of Q^{T}. Homework Equations Qx = λx where x \neq 0 The Attempt at a Solution Qx_{1} = x_{1} for some vector x_{1}...
  48. D

    Linear Algrebra- Orthogonal Vectors

    I am having trouble with these questions- Explain/prove whether: (a) Any set {v1,v2,...vk} of orthogonal vectors in Rn is linearly independent. (b) If there is a vector v in Rn and scalar c in R, we have ||cv|| = c||v|| (c) for any vectors u, v in Rn, ||u+v||^2 + ||u-v||^2 = 2 ||u||^2 +...
  49. V

    A problem on finding orthogonal basis and projection

    Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...
  50. V

    A problem on finding orthogonal basis and projection

    Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1] a) Find an orthogonal basis for span = {x, x^2, x^3} b) Project the function y = 3(x+x^2) onto this basis. --------------------------------------------------------- I know the...
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