Vector Space

The vector space is an idea that can be used 

to study many topics at one time by studying the common properties ( the additive properties and scalar multiplication properties ) that they satisfy.

The real coordinate spaces

When the row or column distinction is irrelevant, or when it is clear from the context, we will use the common symbol Rn to designate a coordinate space.

the trivial subspace

Given a vector space V, the set Z = {0} containing only the zero vector is a subspace of V because (A1) and (M1) are trivially satisfied. 

Non-subspace

straight lines through the origin in R2 are subspaces, but what about straight lines not through the origin? No—they cannot be subspaces because subspaces must contain the zero vector (i.e., they must pass through the origin).

+

the curved line is not a subspace of R2

Consequently, the only proper subspaces of R2 are the trivial subspace and lines through the origin.

Lines and surfaces in R3 thathave curvature cannot be subspaces So the only proper subspaces of R3 are the trivial subspace, linesthrough the origin, and planes through the origin.

Span(S)

Notice that span (S) is a subspace of V

Spanning Sets

Sum of Subspaces

The sum X + Y is again a subspace of V. (4.1.1)