- #1
Dead Boss
- 150
- 1
Hi,
I've been trying to prove that every vector space has a basis.
So starting from the axioms of vector space I defined linear independence and span and then defined basis to be linear independent set that spans the space. I was trying to figure out a direct way to prove the existence of basis, but the best I managed is proof by construction and I'm not sure whether it's valid or not.
Step 1: Pick an arbitrary vector [itex]v[/itex] from [itex]V^i[/itex], then [itex]S^{i+1} := S^i \cup \{v\}[/itex]
Step 2: [itex]V^{i+1} := V^{i} - span(S^{i+1})[/itex]
Starting with [itex]V^1[/itex] as the vector space in question and [itex]S^1[/itex] as empty set, there are two possibilities:
A. After a finite number of steps [itex]V^i = \{\}[/itex], then it can be shown that [itex]S^i[/itex] is a basis.
B. There is no finite number of steps after which [itex]V^i = \{\}[/itex]. In this case I have absolutely no idea what happens.
In hindsight I can define finite dimensional vector space to be vector space for which this construct terminates. Then I have proven that every finite dimensional vector space has a basis.
Now the questions:
1) Is this acceptable definition of finite dimensional vector space?
2) Is this valid proof of existence of basis for finite dimensional vector space?
3) Does case B still yield a basis?
4) Is this a good proof/definition?
I've been trying to prove that every vector space has a basis.
So starting from the axioms of vector space I defined linear independence and span and then defined basis to be linear independent set that spans the space. I was trying to figure out a direct way to prove the existence of basis, but the best I managed is proof by construction and I'm not sure whether it's valid or not.
Step 1: Pick an arbitrary vector [itex]v[/itex] from [itex]V^i[/itex], then [itex]S^{i+1} := S^i \cup \{v\}[/itex]
Step 2: [itex]V^{i+1} := V^{i} - span(S^{i+1})[/itex]
Starting with [itex]V^1[/itex] as the vector space in question and [itex]S^1[/itex] as empty set, there are two possibilities:
A. After a finite number of steps [itex]V^i = \{\}[/itex], then it can be shown that [itex]S^i[/itex] is a basis.
B. There is no finite number of steps after which [itex]V^i = \{\}[/itex]. In this case I have absolutely no idea what happens.
In hindsight I can define finite dimensional vector space to be vector space for which this construct terminates. Then I have proven that every finite dimensional vector space has a basis.
Now the questions:
1) Is this acceptable definition of finite dimensional vector space?
2) Is this valid proof of existence of basis for finite dimensional vector space?
3) Does case B still yield a basis?
4) Is this a good proof/definition?
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