Basis, Dimension and Co-ordinates

NOTE: Consider the vector space R2 over R

I. Example
i. B = {(1, 0), (0, 1)} ⊆ R2
(for facts related to B)

basis example

ii. B = {(1, 1), (1, 2)} ⊆ R2
( for facts related to B)

basis example

II. Construction of basis of R2

  • Enter ( , ) to choose a = ( x, y ) ∈ R2 such that a ≠ 0



  • Enter ( , ) to choose b = (𝑥₁, 𝑦₁) ∈ R2
    such that b ∉ span({a})



Basis

  • B = {a, b} = {(x, y), (x1, y1)} is a basis.

III. Checking whether a given nonempty subset B of R2 is a basis or not?

Proceed Stepwise

Step 1 , if 0 ∈ B

Step 2 , if number of elements in B is not 2

Step 3 , otherwise

  B = {( , ), ( , )}

  

NOTE: Try it yourself for the vector space Rn over R for some other values of n