What is the Significance of Subspaces in Linear Algebra?

[Sasor]

Ok, so I understand that a vector space is basically the span of a set of vectors (i.e.) all the possible linear combination vectors of the set of vectors...

I don't understand the concept behind a subspace or why it's useful.

I know the conditions are:

1. 0 vector must exist in the set
2. If you add two vectors in the set together, you should get another vector in the set
3. If you multiply a vector by a scalar, you should get another vector in the set.


Do conditions 2 and 3 combine? In other words, can the conditions be rewritten as

1. 0 vector must exist
2. A linear combination of some vectors gives another vector in the set

?

Also, graphically, what is the subset supposed to mean? It seems like the only way for something to be a subspace of Rn, for example, would be to be the vector space Rn...

Could someone give me an analogy to spark some intuition...because this seems very abstract?

 

[Number Nine]

2. A linear combination of some vectors gives another vector in the set

Yes, this is fine. A subspace is closed under linear combinations of its vectors.

It seems like the only way for something to be a subspace of Rn, for example, would be to be the vector space Rn...

Really? If you take a line through the origin in R³, can you combine vectors in that line to get a vector outside of the line?

 

[Muphrid]

Subspaces can be viewed as geometric objects containing the origin: the point at the origin, a line through the origin, a plane through the origin, etc. Each of these constitutes a subspace of the overall vector space.

 

[mathwonk]

in R^3 a subspace is just a subset that is flat and contains the origin. like a line or plane through the origin.

 

[Sasor]

But what is the importance of containing the origin? Like, what application would having a subspace be good for?

Also, what is a basis for it and how is that used in such an interpretation?

 

[lavinia]

But what is the importance of containing the origin? Like, what application would having a subspace be good for?

Also, what is a basis for it and how is that used in such an interpretation?

A vector space - subspace or not - must always contain the origin. Otherwise it is not a vector space.

Not sure what you mean by "good for".

 

[arildno]

But what is the importance of containing the origin? Like, what application would having a subspace be good for?

If you multiply a vector with the SCALAR 0, what do you get?

 

[Sasor]

If you multiply a vector with the SCALAR 0, what do you get?

0 vector

 

[mathwonk]

it enables you to use algebra to calculate geometric phenomena. the origin is the zero element for the addition. one can also consider more general subsets that are just flat and do not contain the origin, but you can also deal with those as translates of subspaces, so subspaces also help you understand those.

 

[Vargo]

If you want to solve a linear equation of the type AX=0, the solution is a subspace of the domain (its called the kernel of A).

If you want to solve a linear equation of the type AX=b, the solution is not a subspace, but you still have to understand the kernel of A anyway. You still have to understand subspaces.

e.g. y''+y=0 is a linear equation of type 1. It is a vector subspace of the set of twice differentiable functions. In fact, it is a subspace of dimension 2 and it is generated by the elements sin(t) and cos(t). y=Asin(t)+Bcos(t).

y''+y=1 is a linear equation of type 2. The solutions are y= 1+ Asin(t)+Bcos(t). In order to understand these solutions it is first necessary to understand the subspace of solutions to the earlier equation.