What is a Linear Manifold and How Does it Differ from a Subspace?

[junglebeast]

A set of vectors V defines a Euclidean subspace. A subspace contains the zero vector. Now consider augmenting this space so that a constant vector must be added to the linear combination. The resulting space no longer contains the zero vector so it is not a subspace, but it's clearly some kind of space...what do we call this kind of space?

 

[ice109]

i think that's called an affine space

 

[HallsofIvy]

I would call it a "linear manifold".

I X is such a subset of a vector space and x₀ is a specific vector in X, then V= {x- x₀| x is in X} is a subspace and every member of X can be written "v+ x₀" for some v in V.

The set of all solutions to a homogeneous linear differential equation form a subspace of all analytic functions and the set of all solutions to a non-homogeneous linear equation form a linear manifold. That is why can find general solution to a non-homogenous equation by adding any specific solution to the general solution of the associated homogenous equation.