Vector Spaces II : Finite Dimensional Linear Algebra

Example 3. If X ⊆ RN is a vector space then it is a vector subspace of RN . Example 4. R1 is a vector subspace of R2. But the set [−1, 1] is not a vector subspace because it is not closed under either vector addition or scalar multiplication (for example, 1 + 1 = 2 6∈ [−1, 1]). Geometrically, a vector space in RN looks like a line, plane, or higher dimensional analog thereof, through the origin. A key feature of a vector space X ⊆ RN is that X can be characterized by listing only a few of its vectors. The characterization is not unique, except in the trivial case X = {0}. These characterizing vectors are said to span the space.