A subspace is a non-empty subset of a vector space that is closed under vector addition and scalar multiplication. A non-empty subset M of a vector space V is a subspace if and only if for any vectors m1 and m2 in M and scalars α and β, the linear combination αm1 + βm2 is also in M. Conversely, if M is a subspace of V, then M itself is a vector space over the same field and satisfies the properties of closure under addition and scalar multiplication.

1. Subspace
• Let v be a vector space over field F & M be the non-empty subset of vector
space then M is called Subspace . if m is a vector space over field F.
Vector Space
Vector Space
Subspace
V
M

2. Corollary
• A non-empty subset M of V is a subset of V if and only if m1,m2 ∈M and
𝛼 , 𝛽 ∈ 𝐹 𝛼m1 + 𝛽m2 ∈M
Proof
For subset m1,m2 ∈M
𝜶m1 ∈M
let m1,m2 ∈M and 𝛼 , 𝛽 ∈F
such that 𝛼m1 + 𝛽m2 ∈M (1)
by taking 𝛼 = 𝛽 =1 in (1) we have m1,m2 ∈M

3. Corollary
By taking
𝛽=0 in (1), we get 𝜶m1 ∈M
Hence
by above result, we conclude that M is a subspace.
Conversely
Let M be a subspace of vector space V. Then M itself is a vector space over F.

4. Conversely
Hence
The addition and scalar multiplication are closed in M.
i.e.
foe all 𝛼 , 𝛽, m1,m2 ∈M 𝛼m1 ,
𝛽m2 ∈M
𝛼m1 + 𝛽m2 ∈M

5. Linear Combination
• Let V be a vector space over felid F . let Vi∈V , α𝑖 ∈ F
∀ i=1,………….,m,
Then element of the types
i=1
M
𝛼
1
𝑣
1 = 𝛼1v1 + 𝛼2v2+𝛼3v3 + … … … … … … + 𝛼mvm
are called linear Combination.

6. Spanning Set
• Let V be a vector space over field F. let S be a non-empty set of V then the
set of all linear combination of elements of set S is called spanning set of S
and is denoted by
<S>
OR
L (S)
Thus <S>={ 𝑖=1
𝑡
𝛼1 𝑠1 α𝑖 ∈ F, 𝑠1 ∈ S}