# A question about orthgonal/orthonormal basis

• transgalactic
In summary: If it's a line in R3, then it will have infinitely many vectors that are perpendicular to it, but we don't know what the line is, so we can't know what the vectors are! If W is a plane, then it will have one vector that is perpendicular to it, but we don't know what the plane is, so we don't know what the vector is.
transgalactic

http://img232.imageshack.us/my.php?image=img8282ef1.jpg

my problem with this question starts with this W(and the T shape up side down) simbol

it represents a vector which is perpandicular to W

so why are they ask me to find the orthogonal(perpandicular) basis
to that perpandicular to W vector??(its already perpandicular to W)

so my answer should be the vectors of W
but in the answer they extract the vectors
from the formula and look for a vector which is perpandicular
to both vectors of Wif there were only W then i whould exract the vectors of the formula
and using gramm shmit
i would find the orghonormal basis(which includes in itself orthogonality)

but i was ask to find the orthogonal vectors of this W (upsidedown T)

i don't know what is the formula of its vectors??

Last edited:
Can you find vector(s) such that any and all vector(s) orthogonal to W can be expressed as a linear combination of these basis vectors?

transgalactic said:

http://img232.imageshack.us/my.php?image=img8282ef1.jpg

my problem with this question starts with this W(and the T shape up side down) simbol

it represents a vector which is perpandicular to W

so why are they ask me to find the orthogonal(perpandicular) basis
to that perpandicular to W vector??(its already perpandicular to W)
You seem to be interpreting "orthogonal basis" for $W^{\perp}$ as meaning vectors perpendicular to $W^{\perp}$! That's not correct. An "orthogonal basis" for a vector space, V, consists of vectors in V that are perpendicular to on another. For example, if the overall vector space is R3 and W is the z-axis, then $W^{\perp}$ is the xy-plane. An "orthonormal" basis for that is {(1, 0, 0), (0, 1, 0)}.

so my answer should be the vectors of W
but in the answer they extract the vectors
from the formula and look for a vector which is perpandicular
to both vectors of W

if there were only W then i whould exract the vectors of the formula
and using gramm shmit
i would find the orghonormal basis(which includes in itself orthogonality)

but i was ask to find the orthogonal vectors of this W (upsidedown T)

i don't know what is the formula of its vectors??
We can't answer that without knowing precisely what W is.

## 1. What is an orthogonal basis?

An orthogonal basis is a set of vectors in a vector space that are perpendicular to each other, meaning they have a right angle between them. This property allows for easy computation and manipulation of vector operations.

## 2. How is an orthogonal basis different from an orthonormal basis?

An orthonormal basis is a special type of orthogonal basis where each vector has a length of 1, making them not only perpendicular but also normalized. This means that the vectors have a magnitude of 1 and are particularly useful in applications such as signal processing and linear algebra.

## 3. What is the significance of using an orthogonal/orthonormal basis?

Using an orthogonal or orthonormal basis can simplify many mathematical operations, especially in linear algebra. It allows for easier calculation of projections, finding orthogonal complements, and solving systems of equations. Additionally, it can help with data analysis and signal processing.

## 4. How do you determine if a set of vectors form an orthogonal/orthonormal basis?

To determine if a set of vectors form an orthogonal basis, you can use the dot product between each pair of vectors. If the dot product is equal to 0, then the vectors are orthogonal. To determine if a set of vectors form an orthonormal basis, you can also check if each vector has a magnitude of 1 by taking the square root of the sum of squares of its components.

## 5. Can an orthogonal/orthonormal basis be used in any vector space?

Yes, an orthogonal or orthonormal basis can be used in any vector space, as long as the vectors in the basis are linearly independent. However, in certain vector spaces such as Euclidean space, it is more common to use an orthonormal basis due to its additional properties and simplification of operations.

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