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. S

    Bases, Subspaces, Orthogonal Complements and More to Come

    Homework Statement Show that the set W consisting of all vectors in R4 that are orthogonal to both X and Y is a subspace of R4. Here X and Y are vectors such that X = (1001) and Y = (1010). Part b) Find a basis for W. The Attempt at a Solution So I know to satisfy being a...
  2. G

    Show that this field is orthogonal to each vector field.

    Homework Statement If a, b, and c are any three vector fields in locally Minkowskain 4-manifold, show that the field ε_{ijkl}a^{i}b^{k}c^{l} is orthogonal to \vec{a}, \vec{b}, and \vec{c}. Homework Equations The Attempt at a Solution I know I have to show that multiplying the...
  3. DryRun

    Use cross-product to find vector in R^4 that is orthogonal

    Homework Statement http://s2.ipicture.ru/uploads/20111115/ltM3iwGZ.jpg The attempt at a solution Please correct me if I'm wrong in my assumptions: R^4 means that i need to find a vector that exists in 4 dimensions, meaning 4 rows. I am trying desperately to visualise this problem, with 4...
  4. D

    Basis for the orthogonal complement.

    Homework Statement Let W be the plane 3x + 2y - z = 0 in R3. Find a basis for W^{\perp}Homework Equations N/A The Attempt at a Solution Firstly, I take some arbitrary vector u = \begin{bmatrix}a\\b\\c\end{bmatrix} that is in W^{\perp}. Then I note that W can be rewritten in terms of the...
  5. E

    How do I prove that A^B is orthogonal to A?

    If A = (2,-2,1) and B = (2, 0, -1) show by explicit calculation that; i) A^B is orthogonal to A ii) (A^B)^B lies in the same plane as A and B by expressing it as a linear combination of A and B I'm using; A^B = |A||B|sin θ I know that when you do the cross product of two vectors...
  6. E

    How do I prove that A^B is orthogonal to A?

    Homework Statement If A = (2,-2,1) and B = (2, 0, -1) show by explicit calculation that i) A^B is orthogonal to A ii) (A^B)^B lies in the same plane as A and B by expressing it as a linear combination of A and B Homework Equations A^B = |A||B|sin θ The Attempt at a Solution...
  7. L

    The Orthogonal Complement

    Let M and N be two subsets of a hilbert space H. What are orthogonal complements of following sets: 1) The union of M and N. 2) The intersection of M and N.
  8. S

    Question about problem related to orthogonal complements

    Homework Statement If V is the orthogonal complement of W in Rn, is there such a matrix with row space V and nullspace W? Starting with a basis for V, construct such a matrix. The Attempt at a Solution I've been trying to use the fact that V is the left nullspace of the column space of W...
  9. R

    Do the eigenfunctions for the position operator form an orthogonal set?

    Starting with, \hat{X}\psi = x\psi then, x\psi = x\psi \psi = \psi So the eigenfunctions for this operator can equal anything (as long as they keep \hat{X} linear and Hermitian), right? Well, McQuarrie says that "the eigenfunctions of a Hermitian operator are orthogonal", which can be...
  10. T

    Implicit Differentiation and Orthogonal Trajectories

    Homework Statement Q 50: The ellipse 3x2 +2y2 = 5 and y3 = x2 HINT: The curves intersect at (1,1) and (-1,1) Two families of curves are said to be orthogonal trajectories (of each other) if each member of one family is orthogonal to each member of the other family. Show that the families of...
  11. N

    8 orthogonal projection innequality

    8) U=\{x=(x_{1},x_{2},x_{3},x_{4})\in R^{4}|x_{1}+x_{2}+x_{4}=0\} is a subspace of R^{4} v=(2,0,0,1)\in R^{4} find u_{0}\in U so ||u_{0}-v||<||u-v|| how i tried: U=sp\{(-1,1,0,0),(-1,0,0,1),(0,0,1,0)\} i know that the only u_{0} for which this innequality will work is if it will be the...
  12. J

    Proving two circles are orthogonal

    Homework Statement Homework Equations The Attempt at a Solution Here's an image of what I need to show. I know I need to show that the segment from the center of the smaller circle to F forms a right angle with line segment CF. Alternatively I could show that line segment CH forms a right...
  13. L

    Orthogonal derivative of binormal vector.

    Homework Statement Suppose that r(s) defines a curve parametrically with respect to arc length and the r′(s) is nonzero on the curve. Show that dB/ds is orthogonal to both B(s) and T(s). Conclude that there is a scalar function τ(s) such that dB/ds = −τ (s)N . (This function τ is known as...
  14. D

    Fourier Series coefficients, orthogonal?

    Homework Statement Hello. I need help with orthogonality of the Fourier series coefficients. I know you can use the dirac delta function, (or the kronecker function) in the orthogonality relationship. I want to try and see the derivation using complex form rather than sines and cosines...
  15. T

    Finding the Distance Between 2 Lines and One Orthogonal Line

    Homework Statement Let L1 be the line (0,4,5) + (1,2,-1)t. Let L2 be the line (-10,9,17) + (-11,3,1)t. Find the line L passing through and orthogonal to L1 and L2. What is the distance between L1 and L2? Homework Equations Vector Projection Equation: V • W/|W| The Attempt at...
  16. C

    Proof that the legendre polynomials are orthogonal polynomials

    I'm now studying the application of legendre polynomials to numerical integration in the so called gaussian quadrature. There one exploits the fact that an orthogonal polynomial of degree n is orthogonal to all other polynomials of degree less than n with respect to some weight function. For...
  17. G

    Finding the standard equation for a plane orthogonal to two other planes

    Homework Statement let p1 and p2 be planes in R3, with respective equations: x+5y-z=20 and 2x+5y+2z=20 These planes are not parallel. Find the standard equation for the plane that is orthogonal to both of these planes and contains the origin. The Attempt at a Solution I have only managed to...
  18. J

    Mysterious orthogonal complement formulas

    This is the problem: Suppose A\in\mathbb{R}^{n\times k} is some matrix such that its vertical rows are linearly independent, and 1\leq k\leq n. I want to find a matrix B\in\mathbb{R}^{m\times n} such that n-k\leq m\leq n, its elements can nicely be computed from the A, and such that the...
  19. T

    Proof of r(t) and r'(t) orthogonal on a sphere

    Homework Statement if a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t). show that the curve lies on a sphere with center the origin. Homework Equations The Attempt at a Solution I'm not quite sure how to prove this. I...
  20. D

    Inner product orthogonal vectors

    Homework Statement Let R4 have the Euclidean inner product. Find two unit vectors that are orthogonal to the three vectors u = (2, 1, -4, 0) ; v = (-1, -1, 2, 2) ; w = (3, 2, 5, 4) Homework Equations <u, v> = u1v1 + u2v2 + u3v3 + u4v4 = 0 {orthogonal} The Attempt at a Solution...
  21. B

    Orthogonal change of basis preserves symmetry

    Homework Statement Prove that symmetric and antisymmetric matrices remain symmetric and antisymmetric, respectively, under any orthogonal coordinate transformation (orthogonal change of basis): Directly using the definitions of symmetric and antisymmetric matrices and using the orthogonal...
  22. P

    Differential Equation related to orthogonal trajectories

    Homework Statement Find the orthogonal trajectories of the given family of curves: All circles through the points (1,1) and (-1,-1) I have reduced the problem to finding solutions to the following differential equation: y'=\frac{y^2-2xy-x^2+2}{y^2+2xy-x^2-2} Homework Equations...
  23. DryRun

    Is a set of orthogonal basis vectors for a subspace unique?

    Homework Statement Is a set of orthogonal basis vectors for a subspace unique? The attempt at a solution I don't know what this means. Can someone please explain? I managed to find the orthogonal basis vectors and afterwards determining the orthonormal basis vectors, but I'm not sure what the...
  24. DryRun

    Use cross product formula in R^4 to obtain orthogonal vector

    Use cross product formula in R^4 to obtain a vector that is orthogonal to rows of A Please help with first part and check if i answered the questions correctly. The matrix A = 1 4 -1 2 0 1 0 -1 2 9 -2 2 1. Use cross product formula in R^4 to obtain a vector that is...
  25. DryRun

    How to verify that the nullspace is orthogonal to the row space?

    Homework Statement How to verify that the nullspace is orthogonal to the row space of B? I have inserted the screen-shot of the problem below: http://i29.fastpic.ru/big/2011/0918/10/ca341692cc37b831143f5fe32351db10.jpg Homework Equations Nullspace and orthogonality.The Attempt at a Solution I...
  26. N

    Linear Algebra: Orthogonal matrices

    Homework Statement Hi A matrix M has an inverse iff it is of full column and row rank, and row rank = column rank. Since any orthogonal matrix has full column rank, does that imply that non-singular matrices are orthogonal as well? Cheers, Niles.
  27. matqkks

    Orthogonal Matrices: Importance & Benefits

    Why are orthogonal matrices important?
  28. E

    Find the Orthogonal Trajectories For The Family of Curves

    Hello, forum! I'm a newbie here. I've been visiting this site for a while but just recently joined. Anyways, I was wondering if anyone could help with this problem. I can find the orthogonal trajectories, however, this one is killing me because there is a constant. Allow me to type it below...
  29. T

    Possible Magnitudes of Vectors in Orthogonal Configurations

    Homework Statement A. Given unit vectors a, b, c in the x, y-plane such that a · b = b · c = 0, let v = a + b + c; what are the possible values of |v|? B. Repeat, except a, b, and c are unit vectors in 3-space Homework Equations The Attempt at a Solution I have solutions for both that I'm...
  30. M

    Orthogonal Eigenfunctions (Landau Lifshitz)

    I've been reading QM by Landau Lifgarbagez, in which I've come across a statement I can't seem to get my head around. It states (just before equation 3.6): a_n = SUM a_m. INTEGRAL f_m. f_n. dq ( a_n is the nth coefficient, f_m is the mth eigenfunction of an operator, dq is the...
  31. T

    Norms and orthogonal Polynomials

    Homework Statement Thanks very much for reading. I actually have two problems, I hope it's ok to state both of the in the same thread. 1. Let Vn be the space of all functions having the n'th derivitve in the point x0. I've been given the semi-norm (holds all the norm axioms other than ||v|| =...
  32. F

    Find vectors that produce certain orthogonal projection

    I have vector [ tex ] v [ /tex ] produced by an orthogonal projection of vector [ tex ] w [ /tex ] over plane spanned by vectors [ tex ] u_1 [ /tex ] and [ tex ] u_2 [ /tex ], in a three dimensional space. If I know [ tex ] v [ /tex ], [ tex ] u_1 [ /tex ] and [ tex ] u_2 [ /tex ], how could I...
  33. S

    Linear Algebra orthogonal matrix

    Homework Statement Suppose that A is a real n by n matrix which is orthogonal, symmetric, and positive definite. Prove that A is the identity matrix.Homework Equations Orthogonality means A^t=A^{-1}, symmetry means A^t=A, and positive definiteness means x^tAx>0 whenever x is a nonzero...
  34. matqkks

    Orthogonal Basis: Importance & Benefits

    Why is an orthogonal basis important?
  35. D

    Showing the derivative of a vector is orthogonal to the vector

    Homework Statement http://i.imgur.com/6j8W6.jpg I'm trying to understand that example in the text. I can imagine a curve on a sphere having the derivative vector being orthogonal to the position vector. What I don't understand is, how does "if a curve lies on a sphere with center the origin"...
  36. fluidistic

    Linear algebra, orthogonal matrix proof

    Homework Statement Demonstrate that the following propositions hold if A is an nxn real and orthogonal matrix: 1)If \lambda is a real eigenvalue of A then \lambda =1 or -1. 2)If \lambda is a complex eigenvalue of A, the conjugate of \lambda is also an eigenvalue of A. Homework...
  37. E

    Orthogonal and parallel Curves

    The definition of parallel curve is well defined, such that given two curves, they must remain equidistant to each other. For instance y = (x^2) + 4 and y = (x^2) - 8 are parallel curves in a function the maps x to y. These form parabolas whose vertical distance to one another remains...
  38. D

    Proving transpose of orthogonal matrix orthogonal

    Homework Statement Show that if A is orthogonal, then AT is orthogonal. Homework Equations AAT = I The Attempt at a Solution I would go about this by letting A be an orthogonal matrix with a, b, c, d, e, f, g, h, i , j as its entries (I don't know how to draw that here)...but...
  39. J

    Find an orthogonal basis for the subspace of

    Homework Statement ... R4 consisting of all vectors of the form [a+b a c b+c] Homework Equations Gram-Schmidt process, perhaps? The Attempt at a Solution Not sure how to approach this one. Helpful hint?
  40. Char. Limit

    Question about orthogonal functions

    All right, so I was wondering... I took a look at generating orthogonal functions (over an interval), and say I have these four: \frac{1}{\sqrt{3}} \frac{5}{3} - \frac{2}{3} x \frac{11}{3} \sqrt{\frac{5}{3}} - \frac{10}{3} \sqrt{\frac{5}{3}} x + \frac{2}{3} \sqrt{\frac{5}{3}} x^2...
  41. D

    Orthogonal Basis in 3d is flipping.

    I'm trying to create a circle in 3D based off of 4 inputs. Position1 Position2 LineLength1 LineLength2 The lines start at the positions, and they meet at their very ends. To do this I've gotten the distance between the points, found the radius of the circle, the position of the center of the...
  42. S

    Hello From wiki''There are a fixed number of orthogonal codes,

    hello From wiki.. ''There are a fixed number of orthogonal codes, timeslots or frequency bands that can be allocated for CDM (Sync CDMA), TDMA, and FDMA systems, which remain underutilized (to fail to utilize fully) due to the bursty nature of telephony and packetized data transmissions...
  43. 6

    Student t orthogonal polynomials

    I've just read a paper that references the use of student-t orthogonal polynomials. I understand how the Gauss-Hermite polynomials are derived, however applying the same process to the weight function (1 + t^2/v)^-(v+1)/2 I can't quite get an answer that looks anything like a polynomial...
  44. 0

    Find Orthogonal Compliment to Span({[1 -1 1]T, [1 1 0]})

    Homework Statement Find the orthogonal compliment to Span({[1 -1 1]T, [1 1 0]}) Homework Equations V(transpose)=Null(A) u*v=<u,v>=U(transpose)v The Attempt at a Solution I need help understanding the notation of this problem, I am not sure what my MTX A will look like? I cannot find any...
  45. P

    Relativistic addition of orthogonal velocities

    Homework Statement In a given inertial frame, two particles are shout out simultaneously from a given point, with equal speeds v, in orthogonal directions. What is the speed of each particle relative to the other? Answer: v{(2-\frac{v^{2}}{c^{2}})}^{1/2} Homework Equations Velocity...
  46. H

    Orthogonal Basis for Subspace: Find Solution

    Hi Everyone, I want to ask if I did this problem correctly. Homework Statement Find a orthogonal basis for subspace {[x y z]T|2x-y+z=0} Homework Equations X1= [3 2 -4]T, X2=[4 3 -5]T The Attempt at a Solution Gram-Schmidt: F1=X1= [3 2 -4] F2= X2- ((X2.F1)/||F1||2)F1= [4 3...
  47. ArcanaNoir

    Improper orthogonal matrix plus identity noninvertible?

    Homework Statement If P is an orthogonal matrix with detP = -1, show that I+P has no inverse. (Hint: show that (P^t)(I+P)=(I+P)^t) P^t is P transposed. I is the identity matrix given by PP^t=I a^-1 means inverse a a, b, P and such letters, capital or otherwise, are all matrices, limit to...
  48. F

    Are the Curves 2x^2 + y^2 = 3 and x = y^2 Orthogonal?

    Homework Statement Two curves are said to be orthogonal if their derivatives are opposite reciprocals at the point where the two curves intersect. Are 2x^2 + y^2 =3 and x= y^2 orthogonal?Homework Equations I'm not entirely sure what to put here, but I think one relevant thing is to say that...
  49. B

    Orthogonal Subgroups : on Modules?

    Hi, All: I have seen Orthogonal groups defined in relation to a pair (V,q) , where V is a vector space , and q is a symmetric, bilinear quadratic form. The orthogonal group associated with (V,q) is then the subgroup of GL(V) (invertible linear maps L:V-->V ), i.e., invertible matrices...
  50. S

    Orthogonal Matricies and rotations.

    Hi all, I have been trying to gain a deeper insight into quadratic forms and have realized that my textbook makes the assumption that an orthogonal matrix corresponds to either a rotation and/or reflection when viewed as a linear transformation. The textbook outlines a proof that demonstrates...
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