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

  • Thread starter junglebeast
  • Start date
  • Tags
    Space
In summary, a Euclidean subspace is defined by a set of vectors and includes the zero vector. Augmenting this space with a constant vector creates a type of space known as an affine space. This type of space is also known as a linear manifold and can be represented as V= {x- x0| x is in X}. The set of solutions to a homogeneous linear differential equation is a subspace, while the set of solutions to a non-homogeneous linear equation is a linear manifold. Adding a specific solution to the general solution of the associated homogeneous equation allows us to find the general solution to the non-homogeneous equation.
  • #1
junglebeast
515
2
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?
 
Physics news on Phys.org
  • #2
i think that's called an affine space
 
  • #3
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.
 

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

What is the definition of space?

Space is defined as the vast, empty expanse that exists beyond the Earth's atmosphere. It includes all matter and energy, including stars, planets, galaxies, and the space between them.

What are the different types of space?

There are three main types of space: physical space, psychological space, and outer space. Physical space refers to the three-dimensional area that objects occupy. Psychological space refers to an individual's perception of their surroundings. Outer space is the vast expanse beyond Earth's atmosphere.

How is outer space measured?

Outer space is measured in astronomical units (AU) or light-years. An AU is equal to the average distance between the Earth and the Sun, which is about 93 million miles. A light-year is the distance that light can travel in one year, which is about 5.88 trillion miles.

What is the difference between space and the universe?

While space refers to the vast expanse beyond Earth's atmosphere, the universe refers to everything that exists, including all matter, energy, and space itself. The universe is constantly expanding and is estimated to be about 13.8 billion years old.

Can space be created or destroyed?

According to the law of conservation of mass and energy, space cannot be created or destroyed. It can, however, be altered or expanded due to the expansion of the universe or the gravitational pull of celestial bodies.

Similar threads

I Basis of a Subspace of a Vector Space
Replies
8
Views
2K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
Replies
3
Views
2K
MHB What Is Required to Prove a Subset is a Vector Space?
Replies
3
Views
2K
I About the definition of vector space of infinite dimension
Replies
18
Views
2K
I Trying to get a better understanding of the quotient V/U in linear algebra
Replies
10
Views
2K
I Finding a linear combination to enter a sphere
Replies
3
Views
1K
Not (null T) is a subspace of V?
Replies
6
Views
1K
Definition of an operator in a vector space
Replies
8
Views
2K
A Vector space (no topology) basis
2
Replies
38
Views
6K
I Understanding Z2 Graded Vector Spaces: Definition and Examples
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top