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])?
2026-03-31 21:19:22.1774991962
Multiplying Elements in a Vector Space
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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 $2*(2,3) = (4,6)$.