Linear Algebra - Span and Subspace

[KindofSlow]

Proven in book so I know must be true: "In a vector space, the span of any subset is a subspace."

Based on these following statements, obviously I am misunderstanding something.
If someone could point out all of my errors, wrong statements, etc, I would greatly appreciate it.

V = (set of all vectors spanning R3)
S = subset of V (set of all vectors spanning a plane that does not go through the origin) one example would be x+y+z=4.
The plane is the span of S.
S is a subset.
The plane does not contain the zero vector.
So the plane is not a subspace.

This contradicts what is proven true so I must be wrong (probably in multiple places) but I don't know where I am right and where I am wrong.

Thank you in advance.

 

[pka]

Proven in book so I know must be true: "In a vector space, the span of any subset is a subspace."
Based on these following statements, obviously I am misunderstanding something.
If someone could point out all of my errors, wrong statements, etc, I would greatly appreciate it.
V = (set of all vectors spanning R3)
S = subset of V (set of all vectors spanning a plane that does not go through the origin) one example would be x+y+z=4.
The plane is the span of S.
S is a subset.
The plane does not contain the zero vector.
So the plane is not a subspace.

There are multiple misconceptions in the above.
Now you must understand there are several notations for exactly the same concept.
If \(\displaystyle S\ne\emptyset\) then \(\displaystyle \mathcal{L}(S)\) is the set of all finite linear combinations of elements from \(\displaystyle S\). Some authors even require \(\displaystyle [\mathcal{L}(\emptyset)=\{0\}]\)

Thus \(\displaystyle 0\in\mathcal{L}(S)\) for any \(\displaystyle S\). This is because if \(\displaystyle x\in S\) then \(\displaystyle 0=0\cdot x \in\mathcal{L}(S)\)

So I conclude that you problem is just a huge misunderstanding of what spans are.

 

[Ishuda]

Proven in book so I know must be true: "In a vector space, the span of any subset is a subspace."

Based on these following statements, obviously I am misunderstanding something.
If someone could point out all of my errors, wrong statements, etc, I would greatly appreciate it.

V = (set of all vectors spanning R3)
S = subset of V (set of all vectors spanning a plane that does not go through the origin) one example would be x+y+z=4.
The plane is the span of S.
S is a subset.
The plane does not contain the zero vector.
So the plane is not a subspace.

This contradicts what is proven true so I must be wrong (probably in multiple places) but I don't know where I am right and where I am wrong.

Thank you in advance.

As pla implied, you seem to have a misconception of what a span is [there are actually several different equivalent definitions, one of which pka gave]. Addressing your particular problem
(1)V = (set of all vectors spanning R3)
You definition of V doesn't make sense. One vector can not span R³, it takes three linearly independent vectors to span R³
(2)S = subset of V (set of all vectors spanning a plane that does not go through the origin) one example would be x+y+z=4.
Even had you defined V as something like the set of all sets of vectors which span R³ your example of x+y+z=4 means what? You need to clear up your definitions of V & S.
...

 

[KindofSlow]

Thank you both for the responses, they are helpful.