S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?

In summary: The second paragraph is not pedantic. It is a key point that needs to be addressed in solving this problem.
  • #1
Hall
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##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis?

S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements.

Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)##
Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\}

And similarly for case 2 the basis would be ##A'= \{ (1,0,1), (0,1,0)\} ##.

But how to find the basis for ##S##? A union of A and A' would give us a set whose linear span would go beyond S and hence cannot be a basis for S. Can S have a basis? How do we find it?
 
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  • #2
:welcome:

Perhaps you should check whether ##S## is a vector (sub-) space?
 
  • #3
PeroK said:
:welcome:

Perhaps you should check whether ##S## is a vector (sub-) space?
##(x,x,z)##, ##(x,y,x)## ##\in S##

## (2x, x+y, x+z)## is not in S. S is not a subspace.

Oh yes, thanks.
 
  • #4
I think that's 95% of a proof, but if you want to be very technical, you should prove the thing you wrote down isn't actually of the form (x'x',z') or (x',y',x'). This is typically done by actually picking specific x,y,z and observing that it doesn't work.

Also, in case you didn't realize, you didn't actually pick generic elements of S, since you forced x to be the same in both of them. In some cases you could get unlucky and trick yourself into thinking it is a subspace doing this (obviously it works out ok here)
 
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  • #5
Office_Shredder said:
I think that's 95% of a proof, but if you want to be very technical, you should prove the thing you wrote down isn't actually of the form (x'x',z') or (x',y',x'). This is typically done by actually picking specific x,y,z and observing that it doesn't work.

Also, in case you didn't realize, you didn't actually pick generic elements of S, since you forced x to be the same in both of them. In some cases you could get unlucky and trick yourself into thinking it is a subspace doing this (obviously it works out ok here)
Well, the 2nd paragraph is really pedantic. Thanks.
 
  • #6
Hall said:
Well, the 2nd paragraph is really pedantic.
On the contrary, I think @Office_Shredder makes a cogent point.
 
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1. What is a vector?

A vector is a mathematical object that represents both magnitude and direction. It is commonly used in physics and mathematics to describe quantities such as velocity, force, and displacement.

2. What is a basis?

A basis is a set of vectors that can be used to express any other vector in a given vector space. It is a fundamental concept in linear algebra and is used to describe the structure of vector spaces.

3. How is a basis determined for a vector space?

A basis for a vector space can be determined by finding a set of linearly independent vectors that span the entire space. This means that no vector in the set can be written as a linear combination of the other vectors in the set.

4. How does S relate to the basis?

S is a set of vectors that satisfy a specific condition, in this case, x=y or x=z. The basis for S would consist of vectors that can be used to express any vector in S. In other words, the basis for S would be a subset of S that spans the entire space.

5. Why is determining a basis important?

Determining a basis is important because it allows us to represent any vector in a given vector space in terms of a smaller set of vectors. This can simplify calculations and make it easier to understand the structure of the vector space.

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