- #1
fluidistic
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Homework Statement
In a given basis [itex]\{ e_i \}[/itex] of a vector space, a linear transformation and a given vector of this vector space are respectively determined by [itex]\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0\\ 0&0&5\\ \end{pmatrix}[/itex] and [itex]\begin{pmatrix} 1 \\ 2 \\3 \end{pmatrix}[/itex].
Find the matrix representations of the transformation and the vector in a new basis such that the old one is represented by [itex]e_1 =\begin{pmatrix} 1 \\ 1 \\0 \end{pmatrix}[/itex], [itex]e_2=\begin{pmatrix} 1 \\ -1 \\0 \end{pmatrix}[/itex] and [itex]e_3=\begin{pmatrix} 0 \\ 0 \\1 \end{pmatrix}[/itex].
Homework Equations
Not even sure.
The Attempt at a Solution
I suspect it has to do with eigenvalues since the problem in the assignments is right after an eigenvalue problem. I would not have thought of this otherwise.
It reminds me of 2 similar matrices, say A and B, then A=P^(-1)BP but nothing more than this.
I've found that the given matrix has 3 different eigenvalues, thus 3 eigenvectors (hence non 0 vectors) that are linearly independent (since the 3 eigenvalues are all different) and thus the matrix is similar to a triangular one with the eigenvalues 1, 3 and 5 on the main diagonal.
So now I know that there exist an invertible matrix P such that A=P^(-1)BP. But I'm really stuck at seeing how this can help me.
Any tip's welcome.