Basis & Dimension : Linear Algebra : Degree #Basis #& #Dimension #: #Linear #Algebra #: #Degree

August 03, 2026

For a video explanation, click here 👉https://youtu.be/3ByCs2ILhgo

Basis :

A non-empty set S in a vector space V ( F ) is said to be a basis of V if

i ) S is linearly independent and

ii ) S spans V i.e. L ( S ) = V .

Note :

* In the above definition of a Basis, the second condition L ( S ) = V means every vector in V is a

linear combination of vectors of S.

* For the two dimensional Euclidean Space R², for ( a , b ) in R²,we have

( a , b ) = a ( 1 , 0 ) + b ( 0 , 1 )

i.e. every element in R², is a linear combination of S = { ( 1 , 0 ) , ( 0 , 1 ) }

Hence S = { ( 1 , 0 ) , ( 0 , 1 ) } spans R²,

Since S is clearly linearly independent S is a abasis for R², and this basis is called STANDARD BASIS.

* For the two dimensional Euclidean Space R³, for ( a , b , C ) in R³,we have

( a , b , c ) = a ( 1 , 0 , 0 ) + b ( 0 , 1 , 0 ) + c ( 0 , 0 , 1 )

i.e. every element in R³, is a linear combination of S = { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) }

Hence S = { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) } spans R³ .

Clearly S is linearly independent.

Hence S is a basis for R³and this basis is called the STANDARD BASIS for R³ .

* Similarly For Rⁿ , n-dimension Euclidean space , S = { ( 1 , 0 ,0 , … , 0 ) , ( 0 , 1 , 0 , … , 0 ) , … , ( 0 , 0 , 0 , … , 1 ) } is the STANDARD BASIS for Rⁿ .

DIMENSION :

Definition :

The number of elements of a basis of a vector space is called the Dimension of the vector space.

* For R² , since S = { ( 1 , 0 ) , ( 0 , 1 ) } is the basis for R² and S contains 2 elements, dimension of R² is 2 .

* For R³ since S = { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) } is the basis for R³ and S contains 3 elements, dimension of R³ is 3 .

*Similarly for Rⁿ , S = { ( 1 , 0 ,0 , … , 0 ) , ( 0 , 1 , 0 , … , 0 ) , … , ( 0 , 0 , 0 , … , 1 ) } is the basis for Rⁿ , S contains n elements , the dimension of Rⁿ is n .

* Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2 } forms a basis of V3 ( F ).

* Theorem on Linear Dependence

* linear sum of subspaces

* What is a vector space

* linear span of a set

* L ( S ) = L ( S' )

* Theorem on vector space

* Historical Introduction to Linear Algebra

* Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W

to be a subspace of V are (i) 𝜶 𝞊 W, 𝞫 𝞊 W ⇒ 𝜶 - 𝞫 𝞊 W

(ii) a 𝞊 F , 𝜶 𝞊 W ⇒ a𝜶 𝞊 W.

* Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W