1. let and and that is a Basis for V.
it is known that and also that
prove that there is a another basis to V, C that sustains
2. prove that if: M,N are two nxn matrix so:
then M is similar to N
thanks very much
the point which will easily solve both parts of your problem is this: suppose that and is a linear transformation such that let be such that
then the set is a basis for to prove this, we only need to show that the elements of are linearly independent. so suppose
where are some scalars. if for all we're done. so suppose that let and then we'll have hence:
which gives us because contradiction!
By the linearity ( it's a Linear Transformation):
And now check for :
To prove 2, I will assume you are working on R because you do not mention what field you are working with, it's totally analogous though for the other cases. Take (written as a column, or if you prefer) such that and such that . These exist for and
Consider the transformations and defined by : and
Take the basis of given by (the fact that it's a basis is justified by NonCommAlg's post)
Now note that:
Similarly with
We have:
Hence:
Now let be the standard basis for , that is:
By the change of basis theorem: and: simply because:
then: thus: