Question

a) Determine the matrix of the change of basis of K^{n},
if the old basis is the Standard basis (e_{1}, ...,
e_{n}) and the new base is: (e_{n},
e_{n-1}, ..., e_{1}).

b) Determine the matrix, which describes the changement from the
old basis (e_{1}, e_{2}) of K^{2}, to the
new basis: (e_{1} + e_{2}, e_{1} -
e_{2}).

Answer #1

part(a) the standard basis
={e_{1},e_{2},.......,e_{n}} where
e_{1}=(1,0,0,....0)
e_{2}=(0,1,0,0....0).......e_{n}=(0,0,0.......1)

and the new
basis={e_{n},e_{n-1},......,e_{1})

we have to find the matrix by the given basis

let T be the transformation st
T(e_{1})=e_{n}=(0,0,0.....,1)

T(e_{2})=e_{n-1}=(0,0,0.......,1,0)

...................T(e_{n})=e_{1}=(1,0,0......0)

hence we get the matrix A=

PART(B) here we have given that old basis
={e_{1},e_{2}}={(1,0),(0,1)}

and the new
basis={e_{1}+e_{2},e_{1}-e_{2}}

hence we define here
T(e_{1})=e_{1}+e_{2}

and T(e_{2})=e_{1}-e_{2}

hence we get the matrix A=

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