Vector Section Space

If given a vector space V, then we may form another vector space that is a subset of S of V and use operations on V. Because V is a vector space, summarized operations and scalar multiplication always generate another vector in V. Share the new system uses the S subset of V as a set of course to be vector space, the set must be covered under sums and scalar multiplication operations. That is, the sum of the two elements in S must always be an element of S and the result of a scalar with elements of S must always be an element of S.

For Example   is a subset of R2.

If   is any element of S and α any scalar, then

which is an element of S.
If   and    are two arbitrary elements of S, then the number

It is also an element of S.

Definition,
If S is a non-empty subset of a vector space V, and S satisfies the following conditions,
(i)        α x ϵ S if x ϵ S for any scalar α
(ii)        x + y ϵ S if x ϵ S and y ϵ S
then S is called the subspace of V.
Terms (i) say that S is closed under scalar multiplication. That is, when an element of S is multiplied by a scalar, the result is an element of S.
Terms (ii) say that S is closed under addition. That is, the sum of the two elements of S is always an element of S.
So, if we do calculations using the operations of V and elements from S, we will always produce elements from S. Therefore, the subspace of V is a subset S which is closed under the operations of V.
Suppose that S is the subspace of a vector space V. Using the scalar addition and multiplication operations defined in V, we can form a new mathematical system with S as the appropriate set. It can easily be seen that the eight axioms as a whole will remain valid for this new system. The axioms of A3 and A4 are the result of Theorem 1 and condition (i) of the definition of subspace. The other six axioms are valid for each element of V, so in particular the six axioms are valid for elements of S. So actually each subspace is a vector space.

Example 2.
Suppose S = {(x1, x2, x3)T | x1 = x2}
Then S is the subspace of R3, because αx = (αa, αa, αb)T ϵ S.
If (a, a, b)T and (c, c, d)T are any elements of S, then