I am working on Linear Algebra and the topic is Vector Spaces/Subspaces. The elements of the subspace are defined as ([i], [j]). I am aware that one of the criteria for being a subspace includes that when you multiply an element with another element you get an element in that set. What exactly is meant by 'an element'? Is it ([i], [j]) * ([i_2], [j_2])?
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In a vector space, you can only multiply vectors by scalars. You can’t multiply vectors with each other. – Michael Burr Oct 05 '18 at 00:38

1There is no requirement for vector spaces that you need a defined multiplication *between* vectors. The only requirement with regards to multiplication is that there is a *scalar* multiplication defined for a scalar times a vector. – JMoravitz Oct 05 '18 at 00:38

This answer might be useful: https://math.stackexchange.com/a/969737/155629 – Travis Willse Oct 05 '18 at 00:43

What do the square brackets mean? – user4894 Oct 05 '18 at 00:46
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It is not the case for a subspace that "when you multiply an element with another element you get an element in that set," since a vector space does not require a "multiplication" of elements in the vector space. So, I have no definition of multiplication between $(1,2)$ and $(3,4)$. It does require a definition of multiplication by elements of the ground field (that is, a definition of multiplication by scalars). Closure under scalar multiplication is what is required. So, if $(2,3)$ is in the subspace, so must be $2*(2,3) = (4,6)$.
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