Explaning Orthogonality: Vectors, Subspaces, and Curiosity

  • Thread starter Thread starter talolard
  • Start date Start date
AI Thread Summary
The discussion centers on the concept of orthogonality in vector spaces, specifically the relationship between a vector space V and its subspace U. It clarifies that the equation U ⊕ U⊥ = V indicates that every vector in V can be uniquely expressed as the sum of a vector from U and a vector from its orthogonal complement U⊥, rather than implying that all vectors not in U are orthogonal to U. Examples are provided to illustrate this, such as the x-axis and y-axis in the xy-plane. The confusion arises from misunderstanding the nature of the direct sum, which is not a union of sets. The explanation resolves the initial doubts about the intuitive understanding of these mathematical concepts.
talolard
Messages
119
Reaction score
0

Homework Statement


Hello,
Please pardon me if my terms are off, I'm studying in hebrew and might lacka few of the english words.
Anyway this isn't homewokr but just curiosity.
We learned in class that if V is a vector space and U is a subspace of V then U\oplus U^{\bot}=V
But then it seems to me that this implies that every vector that is in V but not in U is orthogonal to every vector in U. i.e.

This just strikes me as odd and counterintuitive. Is it correct or am I issing something.
Thanks
Tal

The Attempt at a Solution

 
Last edited:
Physics news on Phys.org
I don't find it counterintuitive.. for instance, let the subspace U be some plane in a 3d-V. Then all you need to get V is U and its normal.
And it doesn't say that every vector in V but not in U is orthogonal to U, but rather that with U and the vectors orthogonal to U spans V.
I think the U= some plane in 3d-V-space is the best and most intuitive example i can think of.
 
talolard said:

Homework Statement


Hello,
Please pardon me if my terms are off, I'm studying in hebrew and might lacka few of the english words.
Anyway this isn't homewokr but just curiosity.
We learned in class that if V is a vector space and U is a subspace of V then U\oplus U^{\bot}=V
But then it seems to me that this implies that every vector that is in V but not in U is orthogonal to every vector in U. i.e.
No, that's not what it is saying. U\oplus U^{\bot} is NOT a union of sets. In particular, it does NOT mean that every vector in V is in one or the other of those. It means, rather, that every vector in V is a unique sum of a vector in U and a vector in U^{\perp}.


Suppose V is the xy-plane and U is the x-axis. Then U^{\perp} is the y- axis. Every vector in V, a\vec{i}+ b\vec{j} is a unique sum of a vector in U and a vector in V- here, just a\vec{i} and b\vec{j}, respectively.

A slightly harder example: With V as before, let U= {(x,y)|y= x}, the line y= x. Then U^{\perp} is {(x,y)| y= -x}. Given a vector p\vec{i}+ q\vec{j} how would we write it as a sum of vectors in U and U^{\perp}? Well, any vector in U is of the form a\vec{i}+ a\vec{j} for some number a and every vector in U^{\perp} is of the form b\vec{i}- b\vec{j} for some number b so we would need to find a and b such that (a\vec{i}+ a\vec{j})+ (b\vec{i}- b\vec{j}= p\vec{i}+ q\vec{j}.

That is, (a+ b)\vec{i}+ (a- b)\vec{j}= p\vec{i}+ q\vec{j} so we must have a+ b= p and a- b= q. Adding the two equations, 2a= p+ q so a= (p+ q)/2. Subtracting the two equations, 2b= p- q so b= (p- q)/2.

That is, any vector in R2 can be written, in a unique way, as the sum of a vector in U and a vector in U^{\perp}.

This just strikes me as odd and counterintuitive. Is it correct or am I missing something.
Thanks
Tal

The Attempt at a Solution

 
Thanks for clearing that up.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

MHB Finding a Basis for a Linear Subspace Orthogonal to a Given Point P in R^3
Replies
2
Views
1K
Dimension of orthogonal subspaces sum
Replies
4
Views
2K
A Does Closure Under Multiplication in One Subspace Imply the Same for Another?
Replies
3
Views
2K
B Direct sum decomposition into orthogonal subspaces
Replies
2
Views
1K
I Proof that two linear forms kernels are equal
Replies
4
Views
2K
Is a Vector Orthogonal to a Set Also Orthogonal to Its Span?
Replies
8
Views
2K
Orthogonal Projection Matrices for Points on Subspaces in R^3
Replies
2
Views
5K
I Proving Orthogonality of Vector w/ Schnutz Special Relativity Tensors
Replies
9
Views
2K
I Understanding Z2 Graded Vector Spaces: Definition and Examples
Replies
3
Views
4K
MHB Distance Between a Subspace and a Vector
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top