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Strictly speaking, A Subspace is a Vector Space included in another larger Vector Space.

Definition 2: We say that two subspaces Ui and U2 of V are disjunct. Now we are ready to define what a subspace is. Linear combinations and linear independence A linear combination is a sum of scalar multiples of vectors. Equivalently, the set of all solutions to a system Ax 0 of m homogeneous linear equations in n unknowns. This chapter is a brief survey of basic linear algebra. Subspaces A subset of a vector space is a subspace if it is non-empty and, using the restriction to the subset of the sum and scalar product operations, the subset satisfies the axioms of a vector space.

Yes, this vector set is closed under addition because when any two vectors in the set are added to each other, they produce another vector that will be located inside the vector space too. The null space of an m×n matrix A is a subspace of Rn. Yes, the origin is inside the shaded area on the graph, therefore the vector space contains the zero vector. Given two subspaces U and V, the intersection subspace is spanned by those elements that belong to U U and V V.