1. ## Basis for C^4

The question I'm trying to answer is: find a basis for C^4 with respect to which the matrix of T is an element of L(C^4) is upper triangular, where the matrix of T with respect to the standard basis of C^4 is

1 -1 1 -1
0 0 0 1
1 0 0 1
0 1 0 0

I tried to do row reduction in order to make the matrix into upper triangular and from there get a basis for it but as hard as I try I cannot make this matrix into an upper triangular matrix and so i found this question to be impossible because there is no diagonal matrix. I'm wondering if i'm missing something here which would make the question work or if anyone else finds the same thing?

any help is appreciated

2. You don't really need a diagonal matrix, just an upper triangular matrix.

$\begin{bmatrix}1 & -1 & 1 & -1 & a \\ 0 & 0 & 0 & 1 & b \\ 1 & 0 & 0 & 1 & c\\ 0 & 1 & 0 & 0 & d\end{bmatrix}$