Vector Subspace or Linear Manifold.

[matheinste]

Is there any difference between a vector subspace and a linear manifold.

Paul Halmos in Finite Dimensional Vector Spaces calls them the same thing.

Hamburger and Grimshaw in Linear Trasforms in n Dimensional Vector Space does not use the word subspce at all.

Planet Math says a Linear Manifold is a Linear Subspace possibly moved from the origin ( surely incorrect if it is a vector space ).

I assume the first source is correct, the second just uses a different name and the third is incorrect.

Is this so.

Thanks. Matheeinste

 

[matheinste]

Hello again.

I have just found out that the Planet Math definition is that of an Affine Subspace. So what exactly is the definition of a Linear Manifold

Mateinste

 

[HallsofIvy]

Another example: Thinking of R² as a two dimensional vector space, the straight line (through the origin) y= 2x is a subspace: if (x₀,y₀) and (x₁,y₁) are on that line then (ax₀+bx₁, ay₀+by₁] is also for any numbers, a, b: 2(ax₀+bx₁)= 2ax₀+2bx₁= a(2x₀)+b (2x₁)= ay₀+ by₁.

The straight line y= 2x+ 1 is NOT a subspace: (1, 3) and (2, 5) are both on that line but their sum (3, 8) is NOT. It is however a linear manifold.

Does Halmos say they "are the same thing" or does he say "a vector space is a linear manifold"? (I'll have to look up my copy!)

In general a vector space IS a linear manifold. A linear manifold is a vector space if and only if it contains the 0 vector.

 

[matheinste]

Thankyou Hallsofivy.

In answer to you query Halmos has:

Definition. A non-empty subset U of a vector space V is a subspace or a linear manifold if along with every pair, x and y, of vectors contained in U, every linear combination ax + by is also contained in U.

He adds: a word of warning: along with each vector x, a subspace also contains x - x. Hence if we interpret subspaces as generalized lines and planes we must be careful to consider only lines and planes that pass through the origin.

It appears to me that he is saying that a vector subspace and a linear manifold are the same thing.

I also wonder why the whole of the vector space V of which U is a subspace is not also a linear manifold. Is there any significance for this or am I to understand that it is also a linear manifold as it may itself be a subspace of some higher dimensional space.

Thanks Matheinste.

 

[mathwonk]

what is a glass of "sweet milk" or a "nice knock down argument"? it depends on the local definition. just read the definition in the given book. words mean what the author says they mean, nothing more or less.

 

[Rzz]

Here is what i read in a book on functional analysis.

Let X be a linear space. A subset Y of the space X is called a linear manifold if ax+by $$\in$$ Y for x,y $$\in$$ Y and all numbers a,b such that a+b = 1.

 

[matheinste]

Hello Rzz.

I think this definition has something to do with the convexity of the space.

Matheinste.

 

[morphism]

Although this thread is a few months old, I thought I'd give my two cents.

In functional analysis and operator theory, a linear subspace of a normed space is defined to be a linear manifold (see Rzz's definition above) that is closed in the norm topology. These two definitions exist because we sometimes want to distinguish between the two objects. It turns out that finite dimensional linear manifolds are automatically closed in the norm topology, so this distinction only really occurs in the infinite dimensional setting. This is probably why Halmos (who was an operator theorist) uses the terms "subspace" and "manifold" interchangeably in his book Finite Dimensional Vector Spaces.

 

[Rzz]

Thanx Morphism ...
That really cleared the two notions

 

[michaelamarti]

What is the name of the book by Paul Halmos? Would you recommend it for someone who is just learning linear algebra?

 

[mathwonk]

there is no universal math dictionary. in math its like the conversation in alice and wonderland: namely words mean just what each author chooses for them to mean, nothing else.

 

[matheinste]

Hello michaelamarti.

The name of the book is
Paul Halmos - Finite Dimensional Vector Spaces. Published by Springer. An interesting book but not really for beginners such as myself.

Sheldon Axler - Linear Geometry Done Right. Published by Springer. may be more suitable.

I am sure that the much more experienced people in this forum will point you in the right direction.

Matheinste.

 

[mathwonk]

it is interesting how difficult it is to make a point.

 

[HallsofIvy]

it is interesting how difficult it is to make a point.

A good strong 2 by 4 helps.