Vector Subspaces

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October 15th 2011, 10:29 AM
Hartlw

Re: Vector Subspaces

Quote:

Originally Posted by Plato

That is exactly what I am saying.
Zero is a finite number.

You are confusing the difference between I have two books and I have 2 books.

Two books: "War and Peace," "Uncle Vanya."
2 books: 2, 2, 2, 2, 2,
October 15th 2011, 10:43 AM
Hartlw

Re: Vector Subspaces

A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.
October 15th 2011, 12:05 PM
FernandoRevilla

Re: Vector Subspaces

Quote:

Originally Posted by Hartlw

A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.

Imprecisions of usual language. What about the following?:

Definition We say that the sequence $(x_n)_{n\geq 0}$ has finitely many non zero terms iff there exists $p\in\mathbb{N}$ such that $x_n=0$ for all $n\geq p$ .

This is just what all mathematicians interpret with the expression the sequence $(x_n)_{n\geq 0}$ has finitely many non zero terms.

Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.
October 15th 2011, 12:42 PM
Hartlw

Re: Vector Subspaces

Quote:

Originally Posted by FernandoRevilla

Imprecisions of usual language. What about the following?:

Definition We say that the sequence $(x_n)_{n\geq 0}$ has finitely many non zero terms iff there exists $p\in\mathbb{N}$ such that $x_n=0$ for all $n\geq p$ .

This is just what all mathematicians interpret with the expression the sequence $(x_n)_{n\geq 0}$ has finitely many non zero terms.

Question: Has the zero sequence finitely many non zero elements?. You are going to say yes.

I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.

I have a sequence which has only no terms which aren't zero. The sequence has to be zero.

So W is a subspace because it contains only 0.

* a zero number of non-zero terms. zero is a finite number. Reference Plato
October 15th 2011, 01:01 PM
Hartlw

Re: Vector Subspaces

Quote:

Originally Posted by Hartlw

I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

I have a sequence which has no terms* which aren't zero. The sequence could be 0, but nothing else.

I have a sequence which has only no terms which aren't zero. The sequence has to be zero.

So W is a subspace because it contains only 0.

* a zero number of non-zero terms. zero is a finite number. Reference Plato

Actually, if finite means nore than a zero number of terms, the answer is W is not a subspace because it doesn't contain 0.
If "finite number" is restricted to "zero number," W is a subspace because it only contains 0
October 15th 2011, 02:45 PM
CaptainBlack

Re: Vector Subspaces

Quote:

Originally Posted by Hartlw

This is what I am saying: A sequence of zeros does not have any non-zero entries, so it cannot have finiteley many of them. 0 does not belong to W.

If I have no books do I have a finite numbr of them?

What are you saying?

You won't have a chance to reply to this as I have closed the thread but consider your definition of a finite set, then check how a selection of text or reference books define a finite set.

CB

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