Linear Algebra: UNIT I: Vector Spaces
Definition and Theorem of Subspaces
A subset W of a vector space V over a field F is called a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V.
SUBSPACES
Definition:
A
subset W of a vector space V over a field F is called a subspace of V if W is a
vector space over F with the operations of addition and scalar multiplication
defined on V.
In
any vector space V, {0} and V are subspaces.
Note
It
is not necessary to verify all of the vector space properties to prove that a
subset is a subspace because associative, commutative and four vector space
properties hold true for all vectors in the vector space.
Definition
A
subset W of a vector space is a subspace of V if and only if the following 4
properties hold true.
(i)
x+y ∈ W; x, y ∈ W (W is closed under
addition)
(ii)
c x ∈ W whenever c ∈ F, x ∈ W. (W is closed under
scalar multiplication).
(iii)
W has a zero vector. (ie) 0 ∈
W.
(iv)
Each vector in W has an additive inverse in W.
Theorem 5
Let V be a vector space
and W a subset of V. Then W is a subspace of V if and only if the following 3
conditions hold for the operations defined in V.
(a) 0 ∈ W.
(b) x+y ∈ W whenever x ∈ W and y ∈ W.
(c) cx ∈ W whenever c ∈ F, x ∈ W.
Proof:
Assume
that W is a subspace of V.
W
is a vector space over F with operations defined on V.
⇒ W is additive abelian
group.
⇒ If x, y ∈ W, then x+y ∈ W
If
c ∈ F, x ∈ W then cx ∈ W
To prove the condition
(a)
Let
0 ∈ W. Assume that there
exists a vector 0 ∈
W, such that x+0'=x, for each x ∈
W.
But
we know that x+0=x
x+0=x+0'
0=0'
0 ∈ W. Therefore condition (a) holds.
Conversely
Assume
the conditions (a), (b) and (c) are true.
To prove
that W is subspace of V.
by
condition (b) x+y ∈
W, x, y ∈ W.
(c)
cx ∈ W, C ∈ F, x ∈ W.
(a)
0 ∈ W,
If
x ∈ W and −1 ∈ F then
(−1) x ∈
W by (c).
−x ∈
W.
Additive inverse of each element x lies in W.
W is a
subspace of V.
Linear Algebra: UNIT I: Vector Spaces : Tag: maths, mathematics : - Definition and Theorem of Subspaces
Linear Algebra: UNIT I: Vector Spaces
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Linear Algebra
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