Let V be a
vector space over the field F the linear sum of two subspaces W1 and
W2 of V written as (W1+W2) and is defined as W1+W2={α1+α2:α1єw1,α2єw2} which shows that each element 0f (W1+W2)
is expressible as sum of an element of W1 and an element of W2.
Showing posts with label Vector Space. Show all posts
Showing posts with label Vector Space. Show all posts
Linear Combination & Linear Span
Linear Combination Of Vectors:-
Let V be a
vector space over the field F,then a vector αєV is said to be a linear
combination of vectors α1,α2,α3,………………..αnєV.
If α=a1α1+a2α2+a3α3+…………………………+anαn
where
a1,a2,a3,……………………………….anєF
Direct Sum of a vector space
Let V be a vector space over the field F then
vector space V is said to be direct sum of its subspaces W1 and W2
written as V=W1⊕W2. If each element of V
is uniquely expressible as sum of an element of W1 and an element of W2.
In this case
W1 and W2 are called complementary subspaces.This definition can be extended for
more than two subspaces.
i.e. vector space V is said to be direct sum
of its subspaces W1,W2,W3,………………………………….,Wn if every element αєV can be written in one
and only one way
α=α1+α2+α3+…………………………..+αn
where α1єW1,α2єW2,α3єW3,…………………….αnєWn
Disjoint
subspaces:-
Two
subspaces W1 and W2 of a vector space V over the field F are said to be disjoint if their
intersection(∩) with zero subspace.
i.e. W1 andW2 are disjoint if W1∩W2={0}
Basis, Dimension & Cosets of a vector Space
Basis of
a vector space:-
A
non-empty subset S of a vector space V(F) is said to be its basis if
1-S is
linearly independent.
2-S
generates V i.e. L(S)=V
i.e. each
vector in V is expressible as a linear combination of element of S.
Veactor Space (Theorem-1)
Statement-The
necessary and sufficient condition for a non-empty subset W of vector space V
is that
a,bєF, α,βєW ⇨ aα+bβєW, ∀ a,bєF and α,βєW
Proof- Necessary condition-
Let W be
a subspace of a vector space V over the field F.
∵ W is a vector space over the same field F. So W is a closed under scalar multiplication.
Definition of Vector Space
Let F be an orbitrary field then a
non-empty set V is called a
vector space
over the field F written as V(F) if following axioms are satisfied-
❶-There is defined an internal composition
in V to be denoted additively such that (V,+) is an abelian group.
i.e.
1-Closure axiom:-
If α,β єV
then
α+βєV,
∀ α,βєBV
Subscribe to:
Posts (Atom)