Talk:Linear independence: Difference between revisions
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::If so, we can rephrase both statements. "An eigenvector is an element of the vector space, such that ..." makes perfect sense, it is clear which vector space is meant. With "A linear dependence [...] is an element of the (a) vector space [...]" it is not clear what this vector space is, it is not the vector space of the v<sub>i</sub>-s, nor is it the vector space of linear dependences, as they form no vector space. We have to specify which vector space we mean and say "A linear dependence [...] is an element of the vector space of ''n''-tuples [...]" or shorter "A linear dependence [...] is an ''n''-tuple [...]". |
::If so, we can rephrase both statements. "An eigenvector is an element of the vector space, such that ..." makes perfect sense, it is clear which vector space is meant. With "A linear dependence [...] is an element of the (a) vector space [...]" it is not clear what this vector space is, it is not the vector space of the v<sub>i</sub>-s, nor is it the vector space of linear dependences, as they form no vector space. We have to specify which vector space we mean and say "A linear dependence [...] is an element of the vector space of ''n''-tuples [...]" or shorter "A linear dependence [...] is an ''n''-tuple [...]". |
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::Compare this with the beginning of ''[[quadratic function]]''. "A quadratic function is a vector, ..." would be a bad beginning, as, even though polynomial functions form a vectorspace, a quadratic function is, in the first place, a polynomial function. |
::Compare this with the beginning of ''[[quadratic function]]''. "A quadratic function is a vector, ..." would be a bad beginning, as, even though polynomial functions form a vectorspace, a quadratic function is, in the first place, a polynomial function. [[User:Markus Schmaus|Markus Schmaus]] 15:01, 15 Jun 2005 (UTC) |
Revision as of 15:01, 15 June 2005
I removed the claim that the space of linear dependencies of a a set of vectors is a projectiv space, since I couldn't make sense of it. The space of linear dependencies is a plain vector space, the kernel of the obvious map from Kn to the vector space V. AxelBoldt 15:38 Oct 10, 2002 (UTC)
I've put that comment back, this time with an explanation. Michael Hardy 00:31 Jan 16, 2003 (UTC)
Vector or tuple?
The section "The projective space of linear dependences" currently starts with "A linear dependence among vectors v1, ..., vn is a vector (a1, ..., an)".
In colloquial speech and java "vector" is sometimes used for a tuple of varying length, but in mathematics a vector is an element of a vectorspace and not just any tuple. "A linear dependence [...] is a vector [...]", thus states that linear dependences form a vector space, which is not the case, as the 0 is missing.
Currently a linear dependence is called a vector just as v1 to vn, which can be misunderstood, that a linear dependence is a vector of the same vector space as v1 to vn, but it is not.
A linear dependence is, in the first place, a tuple, or more generally a family. It could have become a vector, if linear dependeces formed a vector space, but they don't, they form a projective space, which is a manifold and linear dependences are points of this manifold. Markus Schmaus 09:45, 15 Jun 2005 (UTC)
- I do not agree that "A linear dependence [...] is a vector [...]" states that linear dependences form a vector space; it only states that they are elements of a vector space. Similarly, one could say "An eigenvector is a vector v such that ...", even though eigenvectors cannot be zero. -- Jitse Niesen 10:37, 15 Jun 2005 (UTC)
- Do we agree that vector simply referes to an element of a vector space?
- If so, we can rephrase both statements. "An eigenvector is an element of the vector space, such that ..." makes perfect sense, it is clear which vector space is meant. With "A linear dependence [...] is an element of the (a) vector space [...]" it is not clear what this vector space is, it is not the vector space of the vi-s, nor is it the vector space of linear dependences, as they form no vector space. We have to specify which vector space we mean and say "A linear dependence [...] is an element of the vector space of n-tuples [...]" or shorter "A linear dependence [...] is an n-tuple [...]".
- Compare this with the beginning of quadratic function. "A quadratic function is a vector, ..." would be a bad beginning, as, even though polynomial functions form a vectorspace, a quadratic function is, in the first place, a polynomial function. Markus Schmaus 15:01, 15 Jun 2005 (UTC)