# Intersect of U and U perpendicular; Orthogonality

• SeannyBoi71
In summary, the conversation focuses on proving that if a vector u is in the intersection of two subspaces, U and U⊥, then u must equal the zero vector. The conversation includes discussions about the dot product, properties of an inner product space, and the geometric interpretation of orthogonal vectors. The final conclusion is that using the fact that the dot product of two orthogonal vectors is equal to 0, it can be shown that u must equal the zero vector in order for it to be in both U and U⊥.
SeannyBoi71

## Homework Statement

Let U be a subspace of ℝn. Show that if u$\in$U$\bigcap$U$\bot$, then u=0.

## The Attempt at a Solution

I know that U$\bot$ will be orthogonal to U, so any vector u in U dotted with any vector in U$\bot$ will equal 0. But that does not necessarily mean that u = 0 so I must prove something else. I don't know where to include the intersect into all of this, we have never used that kind of thing before.

SeannyBoi71 said:

## Homework Statement

Let U be a subspace of ℝn. Show that if u$\in$U$\bigcap$U$\bot$, then u=0.

## The Attempt at a Solution

I know that U$\bot$ will be orthogonal to U, so any vector u in U dotted with any vector in U$\bot$ will equal 0. But that does not necessarily mean that u = 0 so I must prove something else. I don't know where to include the intersect into all of this, we have never used that kind of thing before.

Well, you've almost got it. Think about u^2 and remember that u is in Rn.

u2? u is a vector, so I'm guessing you're insisting I think about u dot u... I'm a little confused how that helps me here

well one way to look at u.u is to think of "the left u" as being in U, and the "right u" as being in U (which we can do since u is in the intersection).

Well if I do it that way then yes I will get the dot product is 0, because those two are orthogonal. I don't see how that makes u = 0, though. I'm missing something here

review the properties a dot product has to have...(positive definite is the property you're after)

SeannyBoi71 said:
u2? u is a vector, so I'm guessing you're insisting I think about u dot u... I'm a little confused how that helps me here

Yes, but also cancel out other cases using the orthogonal relation. Intuitively, if I rotate my u vector to make it orthogonal. What are the point(s) which are shared by both these vectors?

Deveno said:
review the properties a dot product has to have...(positive definite is the property you're after)

Just looked through my textbook and lecture notes and could not find any property like that, do you think you could explain it?

And now I know that u = 0 because it will be the only vector that is in U and U$\bot$ because, well obviously, they are at a 90° angle. I just don't understand how I'm supposed to show it.

Actually come to think of it, can I use the formula $$cosθ = x$\cdot$y/||x|| ||y||$$ and plug in θ=pi/2, showing that u$\cdot$w = 0 , so that u must be equal to the 0 vector? This answer seems a bit trivial...

SeannyBoi71 said:
Just looked through my textbook and lecture notes and could not find any property like that, do you think you could explain it?

And now I know that u = 0 because it will be the only vector that is in U and U$\bot$ because, well obviously, they are at a 90° angle. I just don't understand how I'm supposed to show it.

Actually come to think of it, can I use the formula $$cosθ = x$\cdot$y/||x|| ||y||$$ and plug in θ=pi/2, showing that u$\cdot$w = 0 , so that u must be equal to the 0 vector? This answer seems a bit trivial...

Deveno was probably alluding to the orthogonal inner product space. <x,y>=x.y. An inner product space must never be negative. Also, your method above is correct. Either, x OR y are the 0 vector. But in our example, that is u and u satisfy, u=0 and u=0.
If we write out u=[u1,...,un] and use the fact that u.u=0 and solve for each ui

SeannyBoi71 said:
Just looked through my textbook and lecture notes and could not find any property like that, do you think you could explain it?

And now I know that u = 0 because it will be the only vector that is in U and U$\bot$ because, well obviously, they are at a 90° angle. I just don't understand how I'm supposed to show it.

Actually come to think of it, can I use the formula $$cosθ = x$\cdot$y/||x|| ||y||$$ and plug in θ=pi/2, showing that u$\cdot$w = 0 , so that u must be equal to the 0 vector? This answer seems a bit trivial...

if we are talking about the standard dot product in Rn, then

if u = (u1,u2,...,un)

u.u = (u1)2 + (u2)2+...+(un)2

now, if u.u = 0, we have:

0 u.u = (u1)2 + (u2)2+...+(un)2

one hopes that you know that the square of a real number is non-negative.

the sum of two (or more) non-negative numbers is also ______?

therefore...

*********

you CANNOT conclude that just because u.w = 0 that one of u or w must be 0. for example, (0,-1).(1,0) = 0(1) + (-1)(0) = 0, but neither of these vectors is the 0-vector.

and, geometrically, saying that the 0 vector is at "a right angle" to something, just doesn't make any sense. how do you choose θ?

Deveno said:
and, geometrically, saying that the 0 vector is at "a right angle" to something, just doesn't make any sense. how do you choose θ?

Didn't mean 0 vector is at a right angle. What I meant is that if two vectors are orthogonal (perpendicular), the only point they are going to have in common is 0 i.e. (0,0,0) in R3. And I chose θ to be pi/2 because that is a 90° angle and therefore the two vectors would be orthogonal... Is this logic not correct? I assumed this is what shaon0 was getting at. The thing is that I don't know how to "show" this, like the question asks. I can put it into words but not into equations.

## 1. What is the "Intersect of U and U perpendicular"?

The intersect of U and U perpendicular refers to the point where two lines, U and U perpendicular, intersect at a right angle or 90 degrees. This point is also known as the origin or the zero vector.

## 2. What does it mean for two vectors to be orthogonal or perpendicular?

When two vectors are orthogonal or perpendicular, it means that they are at a 90 degree angle to each other. This can also be seen as the dot product of the two vectors being equal to zero.

## 3. How is orthogonality used in mathematics and science?

Orthogonality is a fundamental concept in mathematics and science, especially in fields such as geometry, linear algebra, and physics. It is used to describe the relationship between two objects or vectors that are at right angles to each other, and it has numerous applications in fields such as engineering, computer graphics, and signal processing.

## 4. Can two vectors be orthogonal if they have different magnitudes?

Yes, two vectors can be orthogonal even if they have different magnitudes. The magnitude of a vector only describes its length, while the direction of the vector is what determines its orthogonality to another vector. As long as the angle between the two vectors is 90 degrees, they are considered orthogonal.

## 5. How is orthogonality related to linear independence?

Orthogonality is closely related to linear independence, as two vectors that are orthogonal to each other are also linearly independent. This means that one vector cannot be written as a linear combination of the other, and they are considered to be completely independent of each other in terms of their direction and magnitude.

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