About vector space and subspaces

[applechu]

Hi:
I am a newbie to linear algebra; I have a problem about
vector space and subspaces. How to distinguish these two
subject. what I know from books is subspace is going through
zero, but I still can not figure out what is the difference between
vector space and subspaces, thanks.

 

[DonAntonio]

Hi:
I am a newbie to linear algebra; I have a problem about
vector space and subspaces. How to distinguish these two
subject. what I know from books is subspace is going through
zero, but I still can not figure out what is the difference between
vector space and subspaces, thanks.

Simple: a vector subspace is a vector space in its own, but the particle "sub" indicates that it is a subset of a vector space that

contains it, and they both are vectors spaces wrt the very same operations and, of course, over the same field.

Thus, for example, the set $\,\,\{t(1,0) \,;\,t\in\mathbb{R}\}\,$ is a v. subspace of $\,\mathbb{R}^2\,$, but $\,\mathbb{R}^2\,$ is

not a subspace of $\,\mathbb{R}^3$ is the former as not a subset of the latter.

DonAntonio

 

[theorem4.5.9]

DonAntonio is correct, but I think his example is confusing.

$\mathbb{R}^2$ is not a subspace of $\mathbb{R}^3$ because the former are vectors with two entries, and the latter are vectors with three entries. They are just different animals altogether, and so cannot be subsets of one or another.

However, in practice we tend to think of $\mathbb{R}^2$ as a subset of $\mathbb{R}^3$ with the tacit understanding that every vector in $\mathbb{R}^2$ gets a zero appended to it.

So $[4,2]\in\mathbb{R^2}$ becomes $[4,2,0]\in\mathbb{R^3}$

With this understanding, $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$

I add this not to nit pick or be confusing. I add this because this is how most people see $\mathbb{R}^2$ and it's important to make explicit why it is wrong to do so (like in DonAntonio's post) and what we implicitly do in our intuition.

 

[applechu]

Thanks a lot

 

[blue_raver22]

Hello,

Vector space and subspaces are both fundamental concepts in linear algebra. A vector space is a set of vectors that can be added together and multiplied by scalars to produce new vectors within the same set. This set must also follow certain properties, such as closure under addition and scalar multiplication, and must contain a zero vector.

On the other hand, a subspace is a subset of a vector space that also satisfies the properties of a vector space. This means that a subspace is also closed under addition and scalar multiplication and contains a zero vector. However, a subspace is not necessarily the entire vector space itself.

To distinguish between the two, you can think of a vector space as a larger set that contains all possible combinations of vectors, while a subspace is a smaller set within that vector space that still follows the rules of a vector space. Another way to think about it is that a vector space is the entire universe, while a subspace is a galaxy within that universe.

I hope this helps clarify the difference between vector space and subspaces. Keep studying and practicing, and you will become more comfortable with these concepts. Good luck!