Question

Define vector subspace
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Answer 1

Answer:

In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace[1][2] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace, when the context serves to distinguish it from other types of subspaces

Answer 2

Some important information :

Vector space :

(V , +) be an algebraic structure and (F , + , •) be a field , then V is called a vector space over the field F if the following conditions hold :

1. (V , +) is an abelian group .
2. ku ∈ V ∀ u ∈ V and k ∈ F
3. k(u + v) = ku + kv ∀ u , v ∈ V and k ∈ F .
4. (a + b)u = au + bu ∀ u ∈ V and a , b ∈ F .
5. (ab)u = a(bu) ∀ u ∈ V and a , b ∈ F .
6. 1u = u ∀ u ∈ V where 1 ∈ F is the unity .

♦ Elements of V are called vectors and the lements of F are called scalars .

♦ If V is a vector space over the field F then it is denoted by V(F) .

Answer :

Subspace :

A non empty subset W of the vector space V(F) is said to be a subspace of V if it itself forms a vector space over the same field F .

♦ A non empty subset W of the vector space V(F) is said to be a subspace of V iff

1. x + y ∈ W ∀ x , y ∈ W
2. ax ∈ W ∀ x ∈ W and a ∈ F

♦ A non empty subset W of the vector space V(F) is said to be a subspace of V iff

1. x + y ∈ W ∀ x , y ∈ W
2. ax ∈ W ∀ x ∈ W and a ∈ F

♦ A non empty subset W of the vector space V(F) is said to be a subspace of V iff ax + by ∈ W ∀ x , y ∈ W and a , b ∈ F .

♦ Every vector space V has atleast two subspaces :

1. Zero vector space {0} is a subspace of V .
2. V is a subspace of V .

♦ Subspaces of R² are :

1. {0} where 0 = (0 , 0)
2. Lines passing through the origin (0 , 0)
3. R²

♦ Subspaces of R³ are :

1. {0} where 0 = (0,0,0)
2. Lines passing through the origin (0 , 0 , 0)
3. Planes passing through the origin (0 , 0 , 0)
4. R³

♦ Intersection of two (or more) subspaces of a vector space is again a subspace of that vector space .

♦ Union of two subspaces of a vector space is again a subspace of that vector space iff one of them is contained in other .

♦ The linear sum of two subspaces of a vector space is again a subspace of that vector space .