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!

  • 2
    $\begingroup$ Some (most?) books don't distinguish "$\subset$" from "$\subseteq$", check out your book notation in the Index of Symbols section $\endgroup$ – francescop21 Jul 12 '18 at 13:52
  • $\begingroup$ See subset : symbols. $\endgroup$ – Mauro ALLEGRANZA Jul 12 '18 at 13:53
  • $\begingroup$ Oh really? hahaha that was easy! Thank you very much ! ! $\endgroup$ – Andreas Mastronikolis Jul 12 '18 at 13:56

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.

  • $\begingroup$ Thank you ! The answer was helpful ! $\endgroup$ – Andreas Mastronikolis Jul 12 '18 at 14:07
  • $\begingroup$ You are welcome. Also, be comfortable with either notation, this will be helpful for you. Usually, in every (undergraduate) mathematics text the notation is given at the starting of the book, so you can look this up and see what system the author is following. $\endgroup$ – астон вілла олоф мэллбэрг Jul 12 '18 at 14:09

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