How to find a vector that is perpendicular to every vector in a linear subspace?

In summary, the person is seeking help with a question about finding a vector perpendicular to every vector in a linear subspace described by x-y+z=0. They are unsure how to use the cross product since no vectors are given. The first step would be to identify the form of the vectors in the subspace, which is <x, y, z> in R3 with the condition x-y+z=0. They are also wondering about the geometric object represented by this subspace.
  • #1
ohjeezus1
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Homework Statement


Hi, i don't know if you can help me but i am currently studying for my finals and i have come across a question which i am very confused about. i have looked it up in books but there seems to be no answer there. the question is Write down a vector of length 1 that is perpendicular to every vector in the linear subspace of r3 described by x-y+z=0. If you could help me i would be very greatful! thank you.
I know that to find a normal vector which is perpendicular to another vector you use the cross product but i do not see how this will benefit me in this question as i am not given any vectors!
 
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  • #2
A first step would be to find out which subspace we're talking about, i.e. write down what form the vectors of this subspace take.
 
  • #3
The subspace is {<x, y, z> in R3 | x - y + z = 0}. What sort of a geometric object is this subspace?
 

1. What is a linear subspace?

A linear subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. In simpler terms, it is a set of vectors that can be combined and scaled to create any other vector in the space.

2. Why is it important to find a vector perpendicular to every vector in a linear subspace?

Finding a vector that is perpendicular to every vector in a linear subspace is important because it allows us to define a new basis for the space. This can simplify calculations and make it easier to understand the properties of the subspace.

3. How can I find a vector that is perpendicular to every vector in a linear subspace?

To find such a vector, you can use the Gram-Schmidt process. This involves finding a basis for the subspace, and then using the orthogonalization procedure to create a new basis where all vectors are perpendicular to each other.

4. Can there be more than one vector that is perpendicular to every vector in a linear subspace?

Yes, there can be multiple vectors that are perpendicular to every vector in a linear subspace. In fact, any vector that is a linear combination of the original basis vectors will also be perpendicular to every vector in the subspace.

5. Is the vector that is perpendicular to every vector in a linear subspace unique?

No, the vector that is perpendicular to every vector in a linear subspace is not unique. As mentioned before, there can be multiple vectors that satisfy this condition. However, the direction of the vector will be unique, as it will always be perpendicular to the subspace.

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