I'm reading a book from dover publications called Introduction to LINEAR ALGEBRA by Marvin Marcus and Henryk Minc. It page 6 it says:
If X is any subset of V (a vector space) then $ \langle X \rangle $ will denote the totality of linear combinations of vectors in X. The set $ \langle X \rangle $ is called the space spanned by X. Each element of $ \langle X \rangle $ is contained in V. We designate that by $ \langle X \rangle \subset V $.
My problem arises in the last statement. I was expecting $ \langle X \rangle \subseteq V $ instead of $ \langle X \rangle \subset V $.
Now I know that if we had two sets A and B and the relationship: $$ A \subset B $$ holds (which means that A is a proper subset of B), that means that every member of A is also a member of B ( $A \subseteq B$) AND $ A \neq B$.
Some further thoughts:
Suppose $ V_3(\mathbb R) $ be a vector space. The set $ I = \{ (1,0,0) , (0,1,0), (0,0,1) \} = \{ \hat i , \hat j , \hat k \} $ is clearly a proper subset of our vector space. So $ I \subset V_3(\mathbb R) $. However we know that I is a basis for $ V_3(\mathbb R) $. And therefore the totality of linear combination of the vectors in I gives you $V_3(\mathbb R) $. Hence: $$ \langle I \rangle = V_3(\mathbb R) $$ which contracdicts the definition from the book. Where am I wrong ? Thanks in Advance!
This is merely a form of notation. There are two that I have predominantly seen :
If $A$ is a subset of $B$ which can possibly be equal to $B$, then we write $A \subset B$. However, if $A$ is a subset of $B$ which cannot be equal to $B$, then we write $A \subsetneq B$, with the bar on the bottom to indicate that $A$ is a proper subset of $B$. This is the more common notation.
The other , which is what you must be used to, is that $A \subset B$ means that $A$ cannot equal $B$, while $A \subseteq B$ means $A$ can equal $B$. This is not as common.
Of course, you are right to realize that there can be equality. However, rather than letting $X$ be a basis for $V$, you could let $X = V$ itself : that itself would show that there is equality in the containment relation.