What is the meaning of subspace and how are they used in linear algebra?

Answer 1

A subspace is a subset of a vector space that also satisfies the conditions of a vector space on its own. For example, in ℜ³, the set of all elements (x, y, 0) forms a subspace of ℜ³, as it respects all the properties of a vector space. To check whether or not a subset of a vector space is a subspace, you need only to check that the subset is closed under the usual operations of addition and scalar multiplication, since all other properties are trivially satisfied by the subset already being part of a vector space.