Question

a) Determine the matrix of the change of basis of Kⁿ, if the old basis is the Standard basis (e₁, ..., e_n) and the new base is: (e_n, e_n-1, ..., e₁).

b) Determine the matrix, which describes the changement from the old basis (e₁, e₂) of K², to the new basis: (e₁ + e₂, e₁ - e₂).

Answer 1

part(a) the standard basis ={e₁,e₂,.......,e_n} where e₁=(1,0,0,....0) e₂=(0,1,0,0....0).......e_n=(0,0,0.......1)

and the new basis={e_n,e_n-1,......,e₁)

we have to find the matrix by the given basis

let T be the transformation st T(e₁)=e_n=(0,0,0.....,1)

T(e₂)=e_n-1=(0,0,0.......,1,0)

...................T(e_n)=e₁=(1,0,0......0)

hence we get the matrix A=

PART(B) here we have given that old basis ={e₁,e₂}={(1,0),(0,1)}

and the new basis={e₁+e₂,e₁-e₂}

hence we define here T(e₁)=e₁+e₂

and T(e₂)=e₁-e₂

hence we get the matrix A=