Currently I'm reading linear algebra books by Leon and Friedberg. In Friedberg's book, to be a subspace, a subset of a vector space should (1). contain zero vector, (2). be closed under scalar multiplication and (3). be closed under vector addition.
But condition (1) is missing in Leon's book.
I think (1) is not necessary since if (2) and (3) holds, then (1) must be true.
Is (1) necessary?
Thanks in advance.
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$\begingroup$Probably it depends on your definition of vector space (i.e.: do you consider $\emptyset$ to be a vector space?) In my opinion, $\emptyset$ is should not be considered a vector space for various reasons, e.g. the fact that $\operatorname{span}\emptyset=\{0\}$, and thus point (1) should be included in the axioms for vector subspaces.
$\endgroup$ 1 $\begingroup$Leon says that a nonempty subset that is closed under scalar multiplication and vector addition is a subspace. It turns out that you can prove that any nonempty subset of a vector space that is closed under scalar multiplication and vector addition always has to contain the zero vector.
Hint: What is zero times a vector? Now use closure under scalar multiplication.
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