Question

Define dimension and a basis of a vector space

Answer 1

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitudeand longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.

Answer 2

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) .

Basis of a vector space :

A set B of vectors in a vector space V is called a basis if all the elements of B are linearly independent and every element of V can be written as a linear combination of elements of B (i.e. B must spans V) .

Dimension of a vector space :

Dimension of a vector space is defined as the number of elements in its basis . The dimension of a vector space V is denoted by dim(V) .

♦ If B = {v₁ , v₂ , v₃ , . . . , vₖ} is a basis of vector space V , then Dimension of V = Cardinality of V , i.e. dim(V) = n(B) = k .

♦ A vector space can have more than one basis .

♦ Every basis of a vector space has the same number of vectors .

Example :

• {(1 , 0) , (0 , 1)} is a basis of the vector space R²(R) , thus its dimension is 2 , i.e. dim(R²(R)) = 2 .
• {1 , x , x²} is a basis of the polynomial space P₃(x) , thus its dimension is 3 , i.e. dim(P₃(x)) = 3 .
• Dim(Rⁿ(R)) = n
• Dim(Pₙ(x)) = n + 1