# 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.

which is an element of S.

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 = {(x

_{1}, x_{2}, x_{3})^{T }| x_{1}= x_{2}}
Then S is the subspace of
R

^{3}, because αx = (αa, αa, αb)^{T}ϵ S.
If (a, a, b)

^{T}and (c, c, d)^{T}are any elements of S, thenSUBSCRIBE TO OUR NEWSLETTER

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