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.
is a subset of R2.
If 
is any element of S and α any scalar, then
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
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
SUBSCRIBE TO OUR NEWSLETTER
0 Response to "Vector Section Space"
Post a Comment