Is it possible to determine a basis for non-subspace spaces?

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Discussion Overview

The discussion revolves around the concept of determining a basis for planes that do not pass through the origin, specifically focusing on the plane described by the equation x − y + z = 2. Participants explore whether a basis can be defined for such non-subspace spaces, the implications of cosets, and the technical definitions of span and basis in vector spaces.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that bases are only defined for vector spaces, implying that without a vector space, the concept of a basis is meaningless.
  • Others introduce the idea of cosets, explaining that a plane not through the origin can be viewed as a coset of a subspace, which can potentially be treated as a vector space under a different set of operations.
  • A participant suggests translating the plane so that it passes through the origin, finding a basis, and then translating back.
  • There is a repeated emphasis that any basis for R^3 must include the zero vector in its span, leading to the conclusion that one cannot exclude the zero vector from the span of a set of vectors.
  • Some participants propose representing the plane in a specific form that involves a linear combination of vectors, questioning whether this aligns with the definition of span.
  • One participant describes a method to find a basis for a plane through the origin that is parallel to the given plane, indicating that any vector in the plane can be expressed as a linear combination of two specific vectors.

Areas of Agreement / Disagreement

Participants generally disagree on the possibility of defining a basis for the plane x − y + z = 2. While some argue that it is impossible due to the technical definition of span, others explore alternative methods and interpretations, indicating that the discussion remains unresolved.

Contextual Notes

The discussion highlights limitations regarding the definitions of vector spaces, bases, and spans, particularly in the context of cosets and planes not passing through the origin. There is an ongoing exploration of how different operations might affect the classification of these sets.

Kubilay Yazoglu
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The title may seem a little confusing and possibly stupid :D

What I mean is like a plane doesn't go through the origin? Can we describe a basis for this? If so, how?
 
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Bases are defined for vector spaces. If you do not have a vector space, the concept of a basis is meaningless.
 
A plane not through the origin is an example of a "coset" in a vector space. A coset of a subspace is the subspace translated by a constant vector. If the coset C of a vector space V doesn't contain the zero vector of V then C can't be a vector space with respect to the operations used in V.

A set of things may not be a vector space under one set of operations and yet can become a vector space under a different set of operations. You can define a set of operations on a coset C in a vector space V that make the coset a vector space. Assume the coset is fomed by adding the vector b to each vector in the subspace S. You can define a new set of operations on C. The basic idea is to un-translate vectors involved in the operations by subtracting b from them. Do the operations on the un-translated vectors with the operations used in V. Then translate the result by adding b to it. A coset with this new set of operations would be a vector space and it could have a basis. However the coset would not be a subspace of the vector space V because subspace of a vector space V must satisfy properties that specify using exactly same operations that are used in V.
 
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Maybe you want to translate so that the vector space goes to the origin, find a base and translate. Then the space is generated by a linear combination plus translation.
 
Stephen Tashi said:
et C of a vector space V doesn't contain the zero vector of V then C can't be a vector space with respect to the operations used in V.
 
Can you determine a basis for the plane x − y + z = 2 that spans only the plane?

This is the original question. I forgot the "only" part. Now, is this possible? Can anyone solve please?
 
Kubilay Yazoglu said:
Can you determine a basis for the plane x − y + z = 2 that spans only the plane?

This is the original question. I forgot the "only" part. Now, is this possible? Can anyone solve please?

You can't solve the problem that you stated. The word "span" has a technical definition. Any basis for R^3 includes the zero vector in it's span. The span of a set of vectors includes the linear combination of those vectors whose coefficients are each the zero scalar. Such a linear combination is equal to the zero vector. So you can't exclude the zero vector from the "span" of a set of vectors.

Perhaps what you want is a representation of the plane in the form (x,y,z) = (x_0,y_0,z_0) + c_1( x_1,y_1,z_1) + c_2(x_2,y_2,z_2). The form of this equation prohibits the coefficient of the vector (x_0,y_0,z_0) from being zero. So the equation does not define the "span" of the set of vectors (x_i, y_i, z_i).
 
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Stephen Tashi said:
You can't solve the problem that you stated. The word "span" has a technical definition. Any basis for R^3 includes the zero vector in it's span. The span of a set of vectors includes the linear combination of those vectors whose coefficients are each the zero scalar. Such a linear combination is equal to the zero vector. So you can't exclude the zero vector from the "span" of a set of vectors.

Perhaps what you want is a representation of the plane in the form (x,y,z) = (x_0,y_0,z_0) + c_1( x_1,y_1,z_1) + c_2(x_2,y_2,z_2). The form of this equation prohibits the coefficient of the vector (x_0,y_0,z_0) from being zero. So the equation does not define the "span" of the set of vectors (x_i, y_i, z_i).
Yes! That is exactly what I've been thinking! But my friends almost convinced me :) Thank you!
 
What you can do, and this is equivalent to what Stephen Tashi said, is look at the plane through the origin, parallel to the given plane. For your example, x- y+ z= 0. That is equivalent to y= x+ z so any vector in it is of the form <x, x+ z, z>= <x, x, 0>+ <0, z, z>= x<1, 1, 0>+ z<0, 1, 1>. That is, any vector can be written as a linear combination of the two vectors < 1, 1, 0> and <0, 1, 1> so they form a basis for the subspace.

Now, one point on the plane x- y+ z= 2 (a plane but NOT a subspace as others have said) is (0, 0, 2) so one vector in the set (not subspace) of vectors representing that plane can be written as the sum of (0, 0, 2) (or any other single vector in the plane) plus a linear combination of <1, 1, 0> and <0, 1, 1>.
 

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