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.

  • 2
    $\begingroup$ Does leon's book include "nonempty" in the definition of a subspace? Condition (1) can be replaced by the condition "the subset is nonempty". $\endgroup$
    – bof
    Jan 5 '14 at 13:10
  • $\begingroup$ Yes leon's definition include 'nonempty' thing. Thanks Adam and bof. $\endgroup$ Jan 5 '14 at 13:17

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.

  • $\begingroup$ I didn't think of empty set. thanks Daniel Robert-Nicoud. $\endgroup$ Jan 5 '14 at 13:19

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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