In a given basis

of a vector space, a linear transformation and a given vector of this space are determinate by

and

(should be a column vector).

Find the matrix representation of the transformation and of the vector in a new basis such that the old one is represented by

,

,

(they should be column vectors).

My attempt: I formed a matrix whose columns are the

,

,

and I found its inverse. Call them

and

.

Then I multiplied

where

is the first matrix I wrote. As result, I obtained the following matrix

which would be the matrix they asked for.

For the vector, I wasn't sure at all... I just multipled

by

and I obtained

which would be the vector they ask for.

Am I right?