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.
Hi everyone,
I would need to get some help on the following question
Let A (m*n)
Let B (m*p)
Let L(A) be the span of the columns of A.
L(A) is orthogonal to L(B) <=> A'B=0
I suppose that the => direction is pretty obvious, since A is in L(A)
and B in is L(B).
Now I am not sure how to...
Let P and Q be two m x m orthogonal projectors.
We show a) ||P-Q||_2 <or eq. 1
b)||P-Q||_2 < 1 implies the ranges of P and Q have equal dimensions.
I think I must use the properties of orthogonal projectors. I guess Range(P) Inters Null(P) = {0} and Range(Q) Inters...
Homework Statement
Find a unit vector that is orthogonal to both i + j and i + k.
I know I can solve this using the cross product of the two. But This chapter is about
dot product and not cross product.
I am not sure how I could go about solving this problem using the properties of...
Homework Statement
i want to get the orthogonal trajectory of the curves of this family
x^2 + y^2=cx
Homework Equations
answer is given as : y^2 + x^2=cy
The Attempt at a Solution
2x + 2yy' = \frac {x^2 +y^2} {x} then y' = \frac{y} {2x} - \frac{x}{y}
let v=y/x ...
How is it the determinant of an orthogonal matrix is \pm1.
Is it:
Suppose Q is an orthogonal matrix \Rightarrow 1 = det(I) = det(QTQ)
= det(QT)det(Q) = ((det(Q))2
and if so, what is it for -1.
Thanks.
The set of orthogonal trajectories for the family indicated by
( x-c)^2 + y^2 = c^2
My work:
y' = -(x-c)/y Since c= ( x^2 + y^2 ) / 2x
plugging back in and doing -1/y' i got
y' = 2xy / ( x^2 - y^2)
Then I am supposed to move the x and y to a side and integrate but i don't...
Homework Statement
Let P be a projection. The definition used is P is a projection if P = PP. Show that ||P|| >=1 with equality if and only if P is orthogonal.
Let ||.|| be the 2-normHomework Equations
P = PP. P is orthogonal if and only if P =P*The Attempt at a Solution
I've proved the...
Homework Statement
Which of the following is the set of orthogonal trajectories for the family indicated by
(x-c)^2 + y^2 = c^2
a). (x-c)^2 + y^2 = c^2
b). (x-c)^2 - y^2 = c^2
c). x^2 + (y-c)^2 = c^2
d). x^2 - (y-c)^2 = c^2
e). None of the above
Homework Equations...
find u X v and show that it is orthogonal to both u and v.
u= 6k
v=-i + 3j + k
http://s763.photobucket.com/albums/xx275/trinhkieu888/?action=view¤t=666.jpg
This is what I got from the picture, but my teacher said that I have one more step to do to show that they are orthogonal, I...
Homework Statement
S1 is in subspace of C^n. P unique orthogonal projector P : C^n -> S1, and x is in range of C^n. Show that the
minimization problem: y in range of S1 so that:
2norm(x-y) = min 2norm(x-z)
where z in range of S1
and
variational problem: y in range of S1 so that...
Homework Statement
P is mxm complex matrix, nonzero, and a projector (P^2=P). Show 2-norm ||P|| >= 1
with equality if and only if P is an orthogonal projector (P=P*)
Homework Equations
Let ||.|| be the 2-norm
The Attempt at a Solution
a. show ||P|| >= 1
let v be in the range...
I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me.
"Properties of the eigenvalue when A is symmetric.
If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times),
orthogonal...
Homework Statement
Let A be an mxn matrix.
a. Prove that the set W of row vectors x in R^m such that xA=0 is a subspace of R^m.
b. Prove that the subspace W in part a. and the column space of A are orthogonal compliments.
Homework Equations
The Attempt at a Solution
a. to...
Homework Statement
Suppose P ∈ L(V) is such that P2 = P. Prove that P is an orthogonal
projection if and only if P is self-adjoint.Homework Equations
The Attempt at a Solution
Let v be a vector in V. Let w be a vector in W and u be a vector in U and let U and W be subspaces of V where dim W+dim...
Suppose X\in\mathbb{R}^{n\times n} is orthogonal. How do you perform the computation of series
\log(X) = (X-1) - \frac{1}{2}(X-1)^2 + \frac{1}{3}(X-1)^3 - \cdots
Elements of individual terms are
((X-1)^n)_{ij} = (-1)^n\delta_{ij} \;+\; n(-1)^{n-1}X_{ij} \;+\; \sum_{k=2}^{n} (-1)^{n-k}...
Homework Statement Given the vectors
u = (2, 0, 1, -4)
v = (2, 3, 0, 1)
Find any unit vector orthogonal to both of them
Homework Equations
I know that two vectors are orthogonal if their dot product is zero...
The Attempt at a Solution
I don't even know how to begin! I know the unit vector...
Homework Statement
Let P\inL(V). If P^2=P, and llPvll<=llvll, prove that P is an orthogonal projection.Homework Equations
The Attempt at a Solution
I think that regarding llPvll<=llvll is redundant. For example, consider P^2=P
and let v be a vector in V. Doesn't P^2=P kind of give it away by...
Aren't all projections orthogonal projections? What I mean is that let's say there
is a vector in 3d space and it gets projected to 2d space. So [1 2 3]--->[1 2 0]
Within the null space is [0 0 3], which is perpendicular to every vector in the x-y plane,
not to mention the inner product of...
Homework Statement
Let W be the subspace spanned by the given column vectors. Find a basis for W perp.
w1= [2 -1 6 3] w2 = [-1 2 -3 -2] w3 = [2 5 6 1]
(these should actually be written as column vectors.
Homework Equations
The Attempt at a Solution
So, I...
Hi, I was just reading about Orthogonal complements.
I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V.
I then proved that the intersection of W and its orthogonal complement equals 0...
Homework Statement
I would like to show that SO3 does not contain any subgroups that are isomorphic to SO2 X SO2.
Homework Equations
I know that any finite subgroup of SO3 must be isomorphic to a cyclic group, a dihedral group, or the group of rotational symmetries of the tetrahedron...
I need some direction.
Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.
I need some direction. I don't have a clue where to start.
Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.
Show that the parabolic coordinates (u,v,\phi) defined by
x=uv \cos{\phi} , y=uv \sin{\phi} , z=\frac{1}{2}(u^2-v^2)
now I am a bit uneasy here because to do this i first need to find the basis vector right?
so if i try and rearrange for u say and then normalise to 1 that will give me...
Homework Statement
Let P2 denote the space of polynomials in k[x] and degree < or = 2. Let f, g exist in P2 such that
f(x) = a2x^2 + a1x + a0
g(x) = b2x^2 + b1x + b0
Define
<f, g> = a0b0 + a1b1 + a2b2
Let f1, f2, f3, f4 be given as below
f1 = x^2 + 3
f2 = 1 - x
f3 = 2x^2 + x + 1
f4 = x +...
Homework Statement
Find an orthogonal basis for the nullspace of the matrix
[2 -2 14]
[0 3 -7]
[0 0 2]
Homework Equations
The Attempt at a Solution
The nullspace is x = [0, 0, 0], so what is the orthogonal basis? It can be anything can't it?
Given a = ( 1, -2, 1), b= ( 0, 1, 2), and c = (-5, -2, 1) determine if {a, b, c} is an orthogonal set. Show support for your answer.
I know that if the dot product of every combination equals zero, the set is orthogonal. No problems here. I do that, and they all equal zero.
a dot b:
1 * 0 +...
I have an assignment question to find an equation of the orthogonal projection onto the XY plane of the curve of intersection of twp particular functions.
If some one knows of a good web page that might explain this to me I would be greatly appreciate it.
regards
Brendan
V is a space of inner muliplication.
W1 and W2 are two subspaces of V, so dimW1<dimW2
prove that there is a vector
0\neq v\epsilon W_2
which is orthogonal to W1
??
W is sub space of R^4 which is defined as
http://img21.imageshack.us/img21/1849/63042233.th.gif
find the system that defines the complements W^\perp of W
i have solved the given system and i got one vector (-1,1,0,0)
so its complement must be of R^3 and each one of the complements...
Let O(n,F_q) be the orthogonal group over finite field F_q. The question is how to calculate the order of the group.
The answer is given in http://en.wikipedia.org/wiki/Orthogonal_group#Over_finite_fields". This seems to be a standard result, but I could not find a proof for this in the basic...
Imagine there is a molecule which consists of several atoms, and for each atom there is an effective orbital, phi_i, which are not orthogonal. Now we want to construct from them a set of orthogonal orbitals, psi_i. Of course there are many ways to do this. Let W be the matrix that realizes our...
Homework Statement
Find all vectors that are perpendicular to (1,4,4,1) and (2,9,8,2)
The Attempt at a Solution
Create matrix A = [[1,4,4,1],[2,9,8,2]]
Set Ax = 0
Reduce by Gauss elimination
Produces basis of (-4,0,1,0) and (-1,0,0,1)
I don't know what the correct solution to...
Homework Statement
Consider a normal operator A
If Rperpendiculara1 is the orthogonal complement to the subspace of eigenvectors of A with eigenvalue a1, show that if y exists in all Rperpendiculara1 then Ay exists in all Rperpendiculara1
The Attempt at a Solution
This could be answered very...
I'm wondering about the action of the complex (special) orthogonal group on \mathbb{C}^n. In the real case, we can use a (real) orthogonal matrix to rotate any (real) vector into some standard vector, say (a,0,0,...,0), where a>0 is equal to the norm of the vector. In other words, the action...
Homework Statement
If w is orthogonal to u and v, then show that w is also orthogonal to span ( u , v )
Homework Equations
two orthogonal vectors have a dot product equalling zero
The Attempt at a Solution
I can see this geometrically in my mind, and I know that w . u = 0 and w ...
Homework Statement
1. If I got a square orthogonal matrix, then if I make up a new matrix from that by rearranging its rows, then will it also be orthogonal?
2. True/false: a square matrix is orthogonal if and only if its determinant is equal to + or - 1
Homework Equations
no...
Hi there!
In order to proof the orthogonal condition aijaik=\delta_{jk} j,k=1,2,3
I write the invariance of the length of a vector in two coordinate systems:
x'ix'i=xixi
Using the linear transformation:
x'i=ai1xi1+ai2xi2+ai3xi3
the first term becomes:
aijaikxjxk
My question is: why...
Homework Statement
I don't understand the difference between an orthogonal complement and it's basis. In this problem: W = [x,y,z]: 2x-y+3z=0 Find w's orthogonal complement and the basis for the orthogonal complement.
The Attempt at a Solution
I did a quick reduced row echelon to [2,-1,3]...
Homework Statement
Prove that an orthogonal transformation T in Rm has 1 as an eigenvalue if the determinant of T equals 1 and m is odd. What can you say if m is even?
The attempt at a solution
I know that I can write Rm as the direct sum of irreducible invariant subspaces W1, W2, ..., Ws...
Greetings,
I'm a little stuck on this algebra question
Find 2 unit vectors orthogonal to v1 = <3,1,1,-1>, v2 = <-1,2,2,0> and v3 = <1,0,2,-1>
I know that this means that if I let z be the vector orthogonal to these 3 then,
z.v1 = z.v2= z.v3 = 0
And that my two vectors is likely +/-...
Homework Statement
http://img55.imageshack.us/img55/8494/67023925dy7.png
The Attempt at a Solution
All functions orthogonal to 1 result in the fact that: \int_a^b f(t)\ \mbox{d}t =0
Now the extra condition is that f must be continous. (because of the intersection).
But...
Homework Statement
Consider the vector space \Renxn over \Re, let S denote the subspace of symmetric matrices, and R denote the subspace of skew-symmetric matrices. For matrices X,Y\in\Renxn define their inner product by <X,Y>=Tr(XTY). Show that, with respect to this inner product,
R=S\bot...
I'm sure there's a simple way of doing this but I just can't think of it.
In R^n, given k<n linearly independent vectors, how do you find a vector that is orthogonal to all given vectors.
I know cross product works for R^3, but what about R^n?
Homework Statement
Given vectors a=(1,-1,0,2,1) b=(3,1,-2,-1,0) and c=(1,5,2,4,-4), which are mutually orthogonal, find a system of linear equations that a vector x must satisfy so it is orthogonal to a, b, and c.Homework Equations
None I think.
The Attempt at a Solution
This is part A of...
I'm curious as to how the following proof is verified. I have toiled over this thing for quite a while, but haven't made any progress. I don't need a step-by-step solution, but I would appreciate any help getting it started:
Given the following:
Q is an orthogonal tensor
e1 is a unit...
hi! I'm new to the forums, and had a question that was more calculus-related than physics. i saw another post similar to this one, but it was incomplete and i couldn't get the answer with the information on it, any chance someone could help me out?
The question is:
"Find a unit vector with a...
Homework Statement
The problem is more of a concept/visual problem, like to find out if it's true or false and why:
all vectors orthogonal to a non zero vector in R^3 are contained in a straight line.
Homework Equations
No equations, just very bad at visualizing in 3-D
The...